Probability
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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

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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} .

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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

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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 welldefined and

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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

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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

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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)=

probability
Let K be a discrete random variable with PMF pK(k)=⎧⎩⎨⎪⎪1/3,2/3,0if k=1,if k=2,otherwise. Conditional on K=1 or 2, random variable Y is exponentially distributed with parameter 1 or 1/2, respectively. Using Bayes' rule,

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

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,

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Suppose that we have a box that contains two coins: A fair coin: P(H)=P(T)=0.5 . A twoheaded 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

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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
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