# 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

25,149 results
1. ## 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

2. ## Math

1. The random variables X and Y are distributed according to the joint PDF fX,Y(x,y) = {ax2,0,if 1≤x≤2 and 0≤y≤x,otherwise. Find the constant a. 2. Determine the marginal PDF fY(y) . (Your answer can be either numerical or an algebraic function of

3. ## probability

The random variables X and Y have the joint PMF pX,Y(x,y)={c⋅(x+y)2,0,if x∈{1,2,4} and y∈{1,3},otherwise. All answers in this problem should be numerical. Find the value of the constant c. c=

4. ## 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

5. ## 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

6. ## 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

7. ## 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

8. ## probability

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)= - unanswered fX(eyy) fX(ln yy) fX(ln y)y none of the above When fX(x) = {1/3,0,if −2

9. ## probability

The random variables X and Y have the joint PMF pX,Y(x,y)={c⋅(x+y)2,0,if x∈{1,2,4} and y∈{1,3},otherwise. All answers in this problem should be numerical. Find the value of the constant c. c= Find P(Y

10. ## Probability & Statistics

The random variable X has a standard normal distribution. Find the PDF of the random variable Y , where: 1. Y = 5X−7 . 2. Y = X2−2X . For y≥−1 ,

11. ## probability

The random variables X and Y are jointly continuous, with a joint PDF of the form fX,Y(x,y)={cxy,if 0≤x≤y≤1 0,,otherwise, where c is a normalizing constant. For x∈[0,0.5], the conditional PDF fX|Y(x|0.5) is of the form ax^b. Find a and b. Your

12. ## 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

13. ## probability

The joint PMF, p X , Y ( x , y ) , of the random variables X and Y is given by the following table: y = 1 4 c 0 2 c 8 c y = 0 3 c 2 c 0 2 c y = − 1 2 c 0 c 4 c x = − 2 x = − 1 x = 0 x = 1

14. ## Math

A random variable X is generated as follows. We flip a coin. With probability p, the result is Heads, and then X is generated according to a PDF fX|H which is uniform on [0,1]. With probability 1−p the result is Tails, and then X is generated according

15. ## Probability

Maximum likelihood estimation The random variables X1,X2,…,Xn are continuous, independent, and distributed according to the Erlang PDF fX(x)= (λ^3*x^2*e^(−λx))/2, for x≥0, where λ is an unknown parameter. Find the maximum likelihood estimate of

16. ## Probability

Random variables X and Y are distributed according to the joint PDF fX,Y(x,y) = {ax,0,if 1≤x≤2 and 0≤y≤x,otherwise.} 1. Find the constant a. 2. Determine the marginal PDF fY(y). (Your answer can be either numerical or algebraic functions of y). For

17. ## 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.

18. ## 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

19. ## 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

20. ## 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

21. ## 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

22. ## 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

23. ## 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

24. ## statistics

A researcher finds that two continuous, random variables of interest, X and Y, have a joint probability density function (pdf) given by: f(x,y)={cxy 0

25. ## Probability

The random variables X and Y have a joint PDF of the form fX,Y(x,y)=c⋅exp{−12(4x2−8x+y2−6y+13)} E[X]= var(X)= E[Y]= var(Y)=

26. ## 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

27. ## 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

28. ## 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

29. ## Probability

The random variable K is geometric with a parameter which is itself a uniform random variable Q on [0,1]. Find the value fQ|K(0.5|1) of the conditional PDF of Q, given that K=1. Hint: Use the result in the last segment.

30. ## 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

31. ## 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

32. ## 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

33. ## probabilty

The random variable X has a PDF of the form fX⁡(x)= 1/x^2,for x≥1, 0,otherwise. Let Y=X2. For y≥1, the PDF of Y it takes the form fY⁡(y)=a / y^b. Find the values of a and b.

34. ## 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

35. ## Math

The random variables Θ and X are described by a joint PDF which is uniform on the triangular set defined by the constraints 0≤x≤1 , 0≤θ≤x . Find the LMS estimate of Θ given that X=x , for x in the range [0,1] . Express your answer in terms of x

36. ## Probability

We are given a stick that extends from 0 to x . Its length, x , is the realization of an exponential random variable X , with mean 1 . We break that stick at a point Y that is uniformly distributed over the interval [0,x] . Find joint PDF fX,Y(x,y) of X

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

Suppose that X and Y are described by a joint PDF which is uniform inside the unit circle, that is, the set of points that satisfy x2+y2≤1 . In particular, the joint PDF takes the value of 1/π on the unit circle. Let Z=X2+Y2−−−−−−−√ ,

39. ## Probability

Suppose that we have three engines, which we turn on at time 0. Each engine will eventually fail, and we model each engine's lifetime as exponentially distributed with parameter λ. The lifetimes of different engines are independent. One of the engines

40. ## 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

41. ## probability

The random variable X has a PDF of the form fX(x)={1x2,0,for x≥1,otherwise. Let Y=X2 . For y≥1 , the PDF of Y it takes the form fY(y)=ayb . Find the values of a and b . a= b=

42. ## 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

43. ## probability

We are given a stick that extends from 0 to x . Its length, x , is the realization of an exponential random variable X , with mean 1 . We break that stick at a point Y that is uniformly distributed over the interval [0,x] . 1. Find joint PDF fX,Y(x,y) of X

44. ## 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

45. ## probability

Random variables X and Y are distributed according to the joint PDF fX,Y(x,y) = {ax,0,if 1≤x≤2 and 0≤y≤x,otherwise. 1 Find the constant a. a= 2 Determine the marginal PDF fY(y). (Your answer can be either numerical or algebraic functions of y). For

46. ## Probability

Let Θ1 and Θ2 be some unobserved Bernoulli random variables and let X be an observation. Conditional on X=x, the posterior joint PMF of Θ1 and Θ2 is given by pΘ1,Θ2∣X(θ1,θ2∣x)= 0.26, if θ1=0,θ2=0, 0.26, if θ1=0,θ2=1, 0.21, if θ1=1,θ2=0,

47. ## 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,

48. ## 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

49. ## Probability

Exercise: MSE As in an earlier exercise, we assume that the random variables Θ and X are described by a joint PDF which is uniform on the triangular set defined by the constraints 0 ≤ x ≤ 1 , 0 ≤ θ ≤ x . a) Find an expression for the conditional

50. ## Math

For t∈R, define the following two functions: f1(t)=12π−−√exp(−max(1,t2)2) and f2(t)=12π−−√exp(−min(1,t2)2). In this problem, we explore whether these functions are valid probability density functions. Determine whether the function f1

51. ## Math

Problem 3. Probability Density Functions 2 points possible (graded, results hidden) For t∈R, define the following two functions: f1(t)=1/√2πexp(−max(1,t^2)^2) and f2(t)=1/√2πexp(−min(1,t^2)^2). In this problem, we explore whether these

52. ## 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

53. ## 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

54. ## 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

55. ## 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

56. ## 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.,

57. ## statistics

A researcher finds that two continuous, random variables of interest, X and Y, have a joint probability density function (pdf) given by: f(x,y)={cxy 0

58. ## probabilities

The random variables X and Y have a joint PDF of the form fX,Y(x,y)=c⋅exp{−12(4x2−8x+y2−6y+13)}.

59. ## 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

60. ## math

Let x1,x2,x3 be random sample with pdf f(x)=exp(-x),x>0 find the joint pdf of y1=x1/x2, y2=x3/(x1+x2),y3=x1+x2 find pdf of y1 Are y1,y2,y3 independent?

61. ## math

let X1,X2,X3 be random sample with pdf f(x)=exp(-x),x>0,find the joint pdf of Y1=X1/X2, Y2=X3/(X1+X2) ,Y3=X1+X2, and find pdf of Y1. Are Y1,Y2,Y3 independent?

62. ## math

let X1,X2,X3 be random sample with pdf f(x)=exp(-x),x>0,find the joint pdf of Y1=X1/X2, Y2=X3/(X1+X2) ,Y3=X1+X2, and find pdf of Y1. Are Y1,Y2,Y3 independent?

63. ## math

Let x1,x2,x3 be random sample with pdf f(x)=exp(-x),x>0 find the joint pdf of y1=x1/x2, y2=x3/(x1+x2),y3=x1+x2 find pdf of y1 Are y1,y2,y3 independent?

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

The random variables X and Y are jointly continuous, with a joint PDF of the form fX,Y(x,y)= {cxy if 0≤x≤y≤1 0 otherwise} where c is a normalizing constant. a) Is it true that fX|Y(2|0.5) is equal to zero? b) Is it true that fX|Y(0.5|2) is equal to

66. ## probability

Paul is vacationing in Monte Carlo. On any given night, he 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

67. ## 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

68. ## Math

As in an earlier exercise, we assume that the random variables Θ and X are described by a joint PDF which is uniform on the triangular set defined by the constraints 0≤x≤1 , 0≤θ≤x . a) Find an expression for the conditional mean squared error of

69. ## 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

70. ## 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.

71. ## 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

72. ## probalitity

Let Θ1 and Θ2 be some unobserved Bernoulli random variables and let X be an observation. Conditional on X=x, the posterior joint PMF of Θ1 and Θ2 is given by pΘ1,Θ2∣X(θ1,θ2∣x)= 0.26, if θ1=0,θ2=0, 0.26, if θ1=0,θ2=1, 0.21, if θ1=1,θ2=0,

73. ## 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=

74. ## 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

75. ## probablity

for 2 indépendant uniform random variables X , Y in [0,3] find conditional pdf f(XY | X = 1/3)

76. ## Probability, Random Variables, and Random Process

A zero-mean Gaussian random process has an auto-correlation function R_XX (τ)={■(13[1-(|τ|⁄6)] |τ|≤6@0 elsewhere)┤ Find the covariance function necessary to specify the joint density of random variables defined at times t_i=2(i-1),i=1,2,…,5.

77. ## Math

Let 𝑋 and 𝑌 be independently random variables, with 𝑋 uniformly distributed on [0,1] and 𝑌 uniformly distributed on [0,2] . Find the PDF 𝑓𝑍(𝑧) of 𝑍=max{𝑋,𝑌} . For 𝑧2 : 𝑓𝑍(𝑧)= For 0≤𝑧≤1 : 𝑓𝑍(𝑧)= For

78. ## 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= -

79. ## stat

let x and y be continuous random variables having joint density f(xy)=a^2 e^-ay, 0

80. ## Statistics

hii. I have a question about pdf and cdf.. I couldn't solve it Please help me :/ 1) the pdf of the random var. X is given by f(x)= (3/8)(x+1)^2 if -1

81. ## statistics

As in an earlier exercise, we assume that the random variables Θ and X are described by a joint PDF which is uniform on the triangular set defined by the constraints 0≤x≤1 , 0≤θ≤x . a) Find an expression for the conditional mean squared error of

82. ## 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

83. ## 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

84. ## 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]=

85. ## 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

86. ## probability

The PDF of exp(X) Let X be a random variable with PDF f_X. Find the PDF of the random variable Y=e^X for each of the following cases: For general f_X, when y>0, f_Y(y)= f_X(ln y) --------- y When f_X(x) = {1/3,0,if −2

87. ## Probability

The random variable K is geometric with a parameter which is itself a uniform random variable Q on [0,1]. Find the value fQ|K(0.5|1) of the conditional PDF of Q, given that K=1. Hint: Use the result in the last segment.

88. ## 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

89. ## Statistics

Let X and Y be independently random variables, with X uniformly distributed on [0,1] and Y uniformly distributed on [0,2] . Find the PDF fZ(z) of Z=max{X,Y} . For z2 : fZ(z)= For 0≤z≤1 : fZ(z)= For 1≤z≤2 : fZ(z)=

90. ## 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

91. ## 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

92. ## Probability

Q1 . 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 .

93. ## Statistics

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

94. ## probability

Maximum likelihood estimation The random variables X1,X2,…,Xn are continuous, independent, and distributed according to the Erlang PDF fX(x)=[λ^3*x^2*e^(−λx)]/2, for x≥0, where λ is an unknown parameter. Find the maximum likelihood estimate of λ,

95. ## mathamatics

The Joint Probability function of two discrete random variable X and Y is given by f(x,y) = c(2x+y), for x= 0,1,2 and Y= 0,1,2,3. Find Joint probability distribution and also evaluate following a) Constant C b) P(x=2, y=1) c) P[(x>=1), y<=2]

96. ## Probability

Consider the experiment of simultaneously tossing a die and a coin. Let X denote the number of heads and Y denote the number of spots showing on the die. a)construct a two-way table for the joint pdf b)Let Z=X+Y . Use the joint pdf to find the distribution

97. ## 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

98. ## 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

99. ## Probability

Consider the following joint PMF of the random variables X and Y: pX,Y(x,y)={1/72⋅(x2+y2),if x∈{1,2,4} and y∈{1,3}, 0, otherwise}. 1. P(Y

100. ## math:Probability

A random variable X is generated as follows. We flip a coin. With probability p , the result is Heads, and then X is generated according to a PDF fX|H which is uniform on [0,1] . With probability 1−p the result is Tails, and then X is generated according