Let N,X1,Y1,X2,Y2,… be independent random variables. The random variable N takes positive integer values and has mean a and variance r. The random variables Xi are independent and identically distributed with mean b and variance s, and the random variables

38,199 results
  1. mathematics, statistics

    You observe k i.i.d. copies of the discrete uniform random variable Xi , which takes values 1 through n with equal probability. Define the random variable M as the maximum of these random variables, M=maxi(Xi) . 1.) Find the probability that M≤m , as a

  2. Probability

    1. Suppose three random variables X , Y , Z have a joint distribution PX,Y,Z(x,y,z)=PX(x)PZ∣X(z∣x)PY∣Z(y∣z). Then, are X and Y independent given Z? 2.Suppose random variables X and Y are independent given Z , then the joint distribution must be of

  3. Statistics

    Let X1,X2,…,Xn be i.i.d. random variables with mean μ and variance σ2 . Denote the sample mean by X¯¯¯¯n=∑ni=1Xin . Assume that n is large enough that the central limit theorem (clt) holds. Find a random variable Z with approximate distribution

  4. statistics

    The random variable W can take on the values of 0, 1, 2, 3, or 4. The expected value of W is 2.8. Which of the following is the best interpretation of the expected value of random variable W? A. A randomly selected value of W must be equal to 2.8. B.The

  5. Probability

    Suppose that we have a box that contains two coins: A fair coin: P(H)=P(T)=0.5 . A two-headed coin: P(H)=1 . A coin is chosen at random from the box, i.e. either coin is chosen with probability 1/2 , and tossed twice. Conditioned on the identity of the

  6. Statistics

    Suppose that X , Y , and Z are independent, with E[X]=E[Y]=E[Z]=2 , and E[X2]=E[Y2]=E[Z2]=5 . Find cov(XY,XZ) . cov(XY,XZ)= Let X be a standard normal random variable. Another random variable is determined as follows. We flip a fair coin (independent from

  7. Economic

    The distance a car travels on a tank of gasoline is a random variable. a. What are the possible values of this random variable? b. Are the Values countable? Explain. c. Is there a finite number of values? Explain. d. Is there random variable discrete or

  8. Math

    For the discrete random variable X, the probability distribution is given by P(X=x)= kx x=1,2,3,4,5 =k(10-x) x=6,7,8,9 Find the value of the constant k E(X) I am lost , it is the bonus question in my homework on random variables so it must be hard. Many

  9. probability

    Let K be a discrete random variable that can take the values 1 , 2 , and 3 , all with equal probability. Suppose that X takes values in [0,1] and that for x in that interval we have fX|K(x|k)=⎧⎩⎨1,2x,3x2,if k=1,if k=2,if k=3. Find the probability

  10. probability

    Problem 2. Continuous Random Variables 2 points possible (graded, results hidden) Let 𝑋 and 𝑌 be independent continuous random variables that are uniformly distributed on (0,1) . Let 𝐻=(𝑋+2)𝑌 . Find the probability 𝐏(ln𝐻≥𝑧) where

  11. Statistics

    Z1,Z2,…,Zn,… is a sequence of random variables that converge in distribution to another random variable Z ; Y1,Y2,…,Yn,… is a sequence of random variables each of which takes value in the interval (0,1) , and which converges in probability to a

  12. Probability

    The joint PMF, pX,Y(x,y), of the random variables X and Y is given by the following table: (see: the science of uncertainty) 1. Find the value of the constant c. c = 0.03571428571428571428 2. Find pX(1). pX(1)= 1/2 3. Consider the random variable Z=X2Y3.

  13. Probability

    Let X1 , X2 , X3 be i.i.d. Binomial random variables with parameters n=2 and p=1/2 . Define two new random variables Y1 =X1−X3, Y2 =X2−X3. We further introduce indicator random variables Zi∈{0,1} with Zi=1 if and only if Yi=0 for i=1,2 . Calculate

  14. Statistics

    Let X be a random variable that takes integer values, with PMF pX(x) . Let Y be another integer-valued random variable and let y be a number. a) Is pX(y) a random variable or a number? b) Is pX(Y) a random variable or a number?

  15. Statistics and Probability

    Let N be a random variable with mean E[N]=m, and Var(N)=v; let A1, A2,… be a sequence of i.i.d random variables, all independent of N, with mean 1 and variance 1; let B1,B2,… be another sequence of i.i.d. random variables, all independent of N and of

  16. probability

    Let X and Y be independent Erlang random variables with common parameter λ and of order m and n, respectively. Is the random variable X+Y Erlang? If yes, enter below its order in terms of m and n using standard notation. If not, enter 0.

  17. Probability

    Question:A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K=5. For K=1,2,3...K, let Xk be a continuous

  18. Math

    Suppose a baseball player had 211 hits in a season. In the given probability distribution, the random variable X represents the number of hits the player obtained in the game. x P(x) 0 0.1879 1 0.4106 2 0.2157 3 0.1174 4 0.0624 5 0.0060 a.) Compute and

  19. probability

    t the discrete random variable X be uniform on {0,1,2} and let the discrete random variable Y be uniform on {3,4}. Assume that X and Y are independent. Find the PMF of X+Y using convolution. Determine the values of the constants a, b, c, and d that appear

  20. Experiment

    the experiment that would allow establishing cause-effect relationship between number of hours worked per week and lower college graduation rates has to have these components: a manipulated independent variable, a dependent variable, control for extraneous

  21. math , probability

    Let X and Y be two independent and identically distributed random variables that take only positive integer values. Their PMF is px (n) = py (n) = 2^-n for every n e N, where N is the set of positive integers. 1. Fix at E N. Find the probability P

  22. math, probability

    13. Exercise: Convergence in probability: a) Suppose that Xn is an exponential random variable with parameter lambda = n. Does the sequence {Xn} converge in probability? b) Suppose that Xn is an exponential random variable with parameter lambda = 1/n. Does

  23. statistics

    Let 𝑋 and 𝑌 be independent positive random variables. Let 𝑍=𝑋/𝑌. In what follows, all occurrences of 𝑥, 𝑦, 𝑧 are assumed to be positive numbers. 1. Suppose that 𝑋 and 𝑌 are discrete, with known PMFs, 𝑝𝑋 and 𝑝𝑌.

  24. Probability

    1.Let 𝑋 and 𝑌 be two binomial random variables: a.If 𝑋 and 𝑌 are independent, then 𝑋+𝑌 is also a binomial random variable b.If 𝑋 and 𝑌 have the same parameters, 𝑛 and 𝑝 , then 𝑋+𝑌 is a binomial random variable c.If 𝑋

  25. Probability

    Let N,X1,Y1,X2,Y2,… be independent random variables. The random variable N takes positive integer values and has mean a and variance r. The random variables Xi are independent and identically distributed with mean b and variance s, and the random

  26. mathematics, probability, statistics

    You observe k i.i.d. copies of the discrete uniform random variable Xi , which takes values 1 through n with equal probability. Define the random variable M as the maximum of these random variables, M=maxi(Xi) . 1.) Find the probability that M≤m , as a

  27. probability

    A fair coin is flipped independently until the first Heads is observed. Let K be the number of Tails observed before the first Heads (note that K is a random variable). For k=0,1,2,…,K, let Xk be a continuous random variable that is uniform over the

  28. Math

    Let N be a positive integer random variable with PMF of the form pN(n)=12⋅n⋅2−n,n=1,2,…. Once we see the numerical value of N , we then draw a random variable K whose (conditional) PMF is uniform on the set {1,2,…,2n} . Write down an expression

  29. mathematics

    For all problems on this page, use the following setup: Let N be a positive integer random variable with PMF of the form pN(n)=(1/2)*(n)*2^(-n),n=1,2,…. Once we see the numerical value of N , we then draw a random variable K whose (conditional) PMF is

  30. probability

    A random experiment of tossing a die twice is performed. Random variable X on this sample space is defined to be the sum of two numbers turning up on the toss. Find the discrete probability distribution for the random variable X and compute the

  31. Probability

    Let X1 , X2 , X3 be i.i.d. Binomial random variables with parameters n=2 and p=1/2 . Define two new random variables Y1 =X1−X3, Y2 =X2−X3. We further introduce indicator random variables Zi∈{0,1} with Zi=1 if and only if Yi=0 for i=1,2 . 1. Calculate

  32. probability

    The vertical coordinate (“height") of an object in free fall is described by an equation of the form x(t)=θ0+θ1t+θ2t2, where θ0, θ1, and θ2 are some parameters and t stands for time. At certain times t1,…,tn, we make noisy observations Y1,…,Yn,

  33. Math

    Searches related to The random variables X1,X2,…,Xn are continuous, independent, and distributed according to the Erlang PDF fX(x)=λ3x2e−λx2, for x≥0, where λ is an unknown parameter. Find the maximum likelihood estimate of λ , based on observed

  34. probablity

    Let X and Y be two independent Poisson random variables, with means λ1 and λ2, respectively. Then, X+Y is a Poisson random variable with mean λ1+λ2. Arguing in a similar way, a Poisson random variable X with parameter t, where t is a positive integer,

  35. Probability

    For each of the following statements, determine whether it is true (meaning, always true) or false (meaning, not always true). Here, we assume all random variables are discrete, and that all expectations are well-defined and finite. Let X and Y be two

  36. Probability

    You observe 𝑘 i.i.d. copies of the discrete uniform random variable 𝑋𝑖 , which takes values 1 through 𝑛 with equal probability. Define the random variable 𝑀 as the maximum of these random variables, 𝑀=max𝑖(𝑋𝑖) 1a. Find the

  37. Probability

    The random variables X1,..,Xn are independent Poisson random variables with a common parameter Lambda . Find the maximum likelihood estimate of Lambda based on observed values x1,...,xn.

  38. Probability

    Let U, V, and W be independent standard normal random variables (that is, independent normal random variables, each with mean 0 and variance 1), and let X=3U+4V and Y=U+W. Give a numerical answer for each part below. You may want to refer to the standard

  39. probability

    This figure below describes the joint PDF of the random variables X and Y. These random variables take values in [0,2] and [0,1], respectively. At x=1, the value of the joint PDF is 1/2. (figure belongs to "the science of uncertainty) 1. Are X and Y

  40. Probability

    For all problems on this page, use the following setup: Let N be a positive integer random variable with PMF of the form pN(n)=12⋅n⋅2−n,n=1,2,…. Once we see the numerical value of N , we then draw a random variable K whose (conditional) PMF is

  41. probability

    et X and Y be independent Erlang random variables with common parameter λ and of order m and n, respectively. Is the random variable X+Y Erlang? If yes, enter below its order in terms of m and n using standard notation. If not, enter 0.

  42. Probability

    Let U, V, and W be independent standard normal random variables (that is, independent normal random variables, each with mean 0 and variance 1), and let X=3U+4V and Y=U+W. Give a numerical answer for each part below. You may want to refer to the standard

  43. ap stats need help

    Continuous Random Variable, I Let X be a random number between 0 and 1 produced by the idealized uniform random number generator described. Find the following probabilities: a.P(0¡ÜX¡Ü0.4) b.P(0.4¡ÜX¡Ü1) c.P(0.3¡ÜX0.5) d.P(0.3(

  44. probability

    Sophia is vacationing in Monte Carlo. On any given night, she takes X dollars to the casino and returns with Y dollars. The random variable X has the PDF shown in the figure. Conditional on X=x , the continuous random variable Y is uniformly distributed

  45. Probability

    For each of the following statements, state whether it is true (meaning, always true) or false (meaning, not always true): 1. Let X and Y be two binomial random variables. (a) If X and Y are independent, then X+Y is also a binomial random variable. (b) If

  46. Statistics/probability

    The random variable X has a binomial distribution with the probability of a success being 0.2 and the number of independent trials is 15. The random variable xbar is the mean of a random sample of 100 values of X. Find P(xbar

  47. Probability

    In the following problem, please select the correct answer. Let X be a non-negative random variable. Then, for any a>0, the Markov inequality takes the form P(X≥a)≤(a^c)E[X^5]. What is the value of c? c= unanswered Suppose that X_1,X_2,⋯ are random

  48. Probability & Statistics

    Exercise: Convergence in probability a) Suppose that Xn is an exponential random variable with parameter λ=n. Does the sequence {Xn} converge in probability? b) Suppose that Xn is an exponential random variable with parameter λ=1/n. Does the sequence

  49. math

    A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K=5 . For k=1,2,…,K , let Xk be a continuous random

  50. Math

    Background: a number, e.g. 2 , can be thought of as a trivial random variable that always takes the value 2 . Let x be a number. Let X be a random variable associated with some probabilistic experiment. a) Is it always true that X+x is a random variable?

  51. Probability

    Let Θ be an unknown random variable that we wish to estimate. It has a prior distribution with mean 1 and variance 2. Let W be a noise term, another unknown random variable with mean 3 and variance 5. Assume that Θ and W are independent. We have two

  52. Probability

    Suppose a random variable X can take any value in the interval [−1,2] and a random variable Y can take any value in the interval [−2,3] . a) The random variable X−Y can take any value in an interval [a,b] . Find the values of a and b : a= b= b) Can

  53. Statistics

    If Y1 is a continuous random variable with a uniform distribution of (0,1) And Y2 is a continuous random variable with a uniform distribution of (0,Y1) Find the joint distribution density function of the two variables. Obviously, we know the marginal

  54. Statistics

    Let X1,…,Xn be i.i.d. Poisson random variables with parameter λ>0 and denote by X¯¯¯¯n their empirical average, X¯¯¯¯n=1n∑i=1nXi. Find two sequences (an)n≥1 and (bn)n≥1 such that an(X¯¯¯¯n−bn) converges in distribution to a standard

  55. Probability

    Let A,B,C be three events and let X=IA, Y=IB, Z=IC be the associated indicator random variables. We already know that X⋅Y is the indicator random variable of the event A∩B. In the same spirit, give an algebraic expression, involving X, Y, Z for the

  56. math

    A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K= 5. For k = 1,2,...,K, let Xk be a continuous random

  57. stats

    Continuous Random Variable, I Let X be a random number between 0 and 1 produced by the idealized uniform random number generator described. Find the following probabilities: a.P(0less than or equal to X less than or equal to 0.4) b.P(0.4 less than or equal

  58. probability

    Problem 4. Gaussian Random Variables Let X be a standard normal random variable. Let Y be a continuous random variable such that fY|X(y|x)=12π−−√exp(−(y+2x)22). Find E[Y|X=x] (as a function of x , in standard notation) and E[Y] . E[Y|X=x]=

  59. probability

    Let 𝑋 and 𝑌 be independent positive random variables. Let 𝑍=𝑋/𝑌 . In what follows, all occurrences of 𝑥 , 𝑦 , 𝑧 are assumed to be positive numbers. 1. Suppose that 𝑋 and 𝑌 are discrete, with known PMFs, 𝑝𝑋 and 𝑝𝑌 .

  60. Probability

    Sophia is vacationing in Monte Carlo. On any given night, she takes X dollars to the casino and returns with Y dollars. The random variable X has the PDF shown in the figure. Conditional on X=x , the continuous random variable Y is uniformly distributed

  61. probablity

    In this problem, you may find it useful to recall the following fact about Poisson random variables. Let X and Y be two independent Poisson random variables, with means λ1 and λ2, respectively. Then, X+Y is a Poisson random variable with mean λ1+λ2.

  62. Probability

    Let N,X1,Y1,X2,Y2,… be independent random variables. The random variable N takes positive integer values and has mean a and variance r. The random variables Xi are independent and identically distributed with mean b and variance s, and the random

  63. probability

    Let N,X1,Y1,X2,Y2,… be independent random variables. The random variable N takes positive integer values and has mean a and variance r. The random variables Xi are independent and identically distributed with mean b and variance s, and the random

  64. probability

    1) Let X and Y be independent continuous random variables that are uniformly distributed on (0,1) . Let H=(X+2)Y . Find the probability P(lnH≥z) where z is a given number that satisfies e^z

  65. probability

    Let 𝑋 and 𝑌 be independent continuous random variables that are uniformly distributed on (0,1) . Let 𝐻=(𝑋+2)𝑌 . Find the probability 𝐏(ln𝐻≥𝑧) where 𝑧 is a given number that satisfies 𝑒^𝑧

  66. probability

    Let X and Y be independent continuous random variables that are uniformly distributed on (0,1). Let H=(X+2)Y. Find the probability P(lnH≥z) where z is a given number that satisfies ez

  67. Probability

    Consider three random variables X, Y, and Z, associated with the same experiment. The random variable X is geometric with parameter p∈(0,1). If X is even, then Y and Z are equal to zero. If X is odd, (Y,Z) is uniformly distributed on the set

  68. Probability

    Let X be a random variable that takes non-zero values in [1,∞), with a PDF of the form fX(x)=⎧⎩⎨cx3 if x≥1, 0,otherwise. Let U be a uniform random variable on [0,2]. Assume that X and U are independent. What is the value of the constant c? c=

  69. probability

    Determine whether each of the following statement is true (i.e., always true) or false (i.e., not always true). 1. Let X be a random variable that takes values between 0 and c only, for some c≥0, so that P(0≤X≤c)=1. Then, var(X)≤c2/4. TRUE 2. X and

  70. Probability

    Let X be a random variable that takes non-zero values in [1,∞), with a PDF of the form fX(x)=⎧⎩⎨cx3,0,if x≥1,otherwise. Let U be a uniform random variable on [0,2]. Assume that X and U are independent. What is the value of the constant c? c= -

  71. Stor

    Here is a simple way to create a random variable X that has mean μ and stan- dard deviation σ: X takes only the two values μ−σ and μ+σ, eachwith probability 0.5. Use the definition of the mean and variance for discrete random variables to show that

  72. probability

    The nth moment of a random variable X is defined to be the expectation E[Xn] of the nth power of X. Recall that a Bernoulli random variable with parameter p is a random variable that takes the value 1 with probability p, and the value 0 with probability

  73. Probability

    The random variable V takes values in the set {0,1} and the random variable W takes values in the set {0,1,2} . Their joint PMF is of the form pV,W(v,w)=c⋅(v+w), where c is some constant, for v and w in their respective ranges, and is zero everywhere

  74. Experiment

    The question is how do I design a basic experiment that would allow us to establish a cause-effect relationship between number of hours worked per week and lower college graduation rates? It must have these components: a manupulated independent variable, a

  75. mathematics

    Applying Linear Functions to a Random Sequence 3 points possible (graded) Let (Zn)n≥1 be a sequence of random variables such that n−−√(Zn−θ)−→−−n→∞(d)Z for some θ∈R and some random variable Z. Let g(x)=5x and define another

  76. social psy

    I want to utilize a true experimental design to study the effects of classical music exposure on the cognitive development of newborns. a- What is an independent variable? b- What is a dependent variable? c-What is random assignment? d-Why is random

  77. Stats

    In each of the four examples listed below, one of the given variables is independent (x) and one of the given variables is dependent (y). Indicate in each case which variable is independent and which variable is dependent. I. Rainfall; Crop yield II. Taxi

  78. math

    A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K=5 . For k=1,2,…,K , let Xk be a continuous random

  79. probabilities

    Let X be a Poisson random variable with µ = EX = 0.4 and let Y be another random such that E[(2Y + 1)2 ] = 10 and E[(Y − 1)2 ] = 4 Consider the random variable Z such that Z = 3X + 4Y + 2. 1. Find E(X2 ), E(Y 2 ) and E(Z). 2. Suppose that X and Y are

  80. science

    what should you ask yourself when looking for an independent variable in an experiment? I would ask whether that variable can be manipulated or not. Here is more info on experimental variables that might be helpful. An independent variable is the potential

  81. Probability

    Let A,B,C be three events, and let X=IA, Y=IB, and Z=IC be the associated indicator random variables. We already know that X⋅Y is the indicator random variable of the event A∩B. In the same spirit, give an algebraic expression, involving X,Yand Z, for

  82. Stats-Probability

    Would really urgently appreciate answers to these questions. Thanks. 3. Suppose that two lotteries each have n possible numbers and the same payoff. In terms of expected gain, is it better to buy two tickets from one of the lotteries or one from each? 4. A

  83. Probability

    Let A,B,C be three events, and let X=Ia,Y=Ib, and Z=Ic be the associated indicator random variables. We already know that X.Y is the indicator random variable of the event A(intersection)B. In the same spirit, give an algebraic expression, involving X,Y

  84. Probability

    Let N be a geometric r.v. with mean 1/p; let A1,A2,… be a sequence of i.i.d. random variables, all independent of N, with mean 1 and variance 1; let B1,B2,… be another sequence of i.i.d. random variable, all independent of N and of A1,A2,…, also with

  85. Probability

    Let Θ be an unknown random variable that we wish to estimate. It has a prior distribution with mean 1 and variance 2. Let W be a noise term, another unknown random variable with mean 3 and variance 5. Assume that Θ and W are independent. We have two

  86. Probability

    This figure below describes the joint PDF of the random variables X and Y. These random variables take values in [0,2] and [0,1], respectively. At x=1, the value of the joint PDF is 1/2. Are X and Y independent? - unanswered Yes No Find fX(x). Express your

  87. Stats

    6. In each of the four examples listed below, one of the given variables is independent (x) and one of the given variables is dependent (y). Indicate in each case which variable is independent and which variable is dependent. I. Rainfall; Crop yield II.

  88. Statisitcs

    Suppose that X and Y are independent discrete random variables and each assumes the values 0,1, and 2 with probability of 1/3 each. Find the frequency function of X+Y.

  89. Probability and statistics

    8. A random variable X takes exactly the 5 values 1, 2,3,4,5, all with same probability. The mean of X is Choose one answer a. 2.5 b. 15 c. 7.5 d. 3

  90. statistics

    Identify the given item as probability distribution, continuous random variable, or discrete random variable. The amount of time that an individual watches television. a. discrete random variable b. probability distribution c. continuous random

  91. Probability

    Suppose that we have a box that contains two coins: A fair coin: P(H)=P(T)=0.5 . A two-headed coin: P(H)=1 . A coin is chosen at random from the box, i.e. either coin is chosen with probability 1/2 , and tossed twice. Conditioned on the identity of the

  92. Statistics

    You would like to determine the percentage of coffee drinkers in your university, and collected the following binary data set from random students on campus, 1 for coffee drinker and 0 for otherwise: 0,0,0,1,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0,1. Let Yi

  93. probability theory

    Let X, Y, Z, be independent discrete random variables. Let A= X(Y+Z) and B= XY With A, B, X, defined as before, determine wheter the folllowing statements are true or false. 1. A and B are independent 2. A and B are conditionally independent, given X = 0.

  94. math

    8 divided by 3 plus 6= You asked 8 divided by 3 plus 6= (8/3) + 6 = 2(2/3) + 6 or 8/3 + 18/3 = ??/3 I don't understand how to describe pairs of related variables what's the difference between dependent & independent variables I'm not what class your

  95. probability

    This figure below describes the joint PDF of the random variables X and Y. These random variables take values over the unit disk. Over the upper half of the disk (i.e., when y>0), the value of the joint PDF is 3a, and over the lower half of the disk (i.e.,

  96. Math, statistics

    Problem 1: The PDF of exp(X) (6/6 points) Let X be a random variable with PDF fX. Find the PDF of the random variable Y=eX for each of the following cases: For general fX, when y>0, fY(y)= Solution: f_x(ln(y))/y When fX(x) = {1/3,0,if −2

  97. statics

    A bag contains 4 balls, each of which have a positive integer written on it. Let X be the random variable of the value written on a single ball drawn from the bucket. If E[X]=8, what is the maximum possible value of E[X^2]?

  98. Math

    Random variables X and Y are both normally distributed with mean 100 and standard deviation 4. It is known that random variable X+Y is also a normal distribution. a. What is the mean of X+Y? b. What is the standard deviation of X+Y? I see that the mean is

  99. ap stats

    Continuous Random Variable, I Let X be a random number between 0 and 1 produced by the idealized uniform random number generator described. Find the following probabilities: a.P(0less than or equal to X less than or equal to 0.4) b.P(0.4 less than or equal

  100. Statistics

    In a population, heights of males are normally distributed with u=180 cm and sigma^2=16 cm^2, while the heights of females are normally distributed with u=170 cm and sigma^2= 25 cm^2. a) One random male and one random female are selected from the

Pages

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 10
  11. 11
  12. 12
  13. 13
  14. 14
  15. 15
  16. 16
  17. 17
  18. 18
  19. 19
  20. 20