(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.

Q: For two random variables X and Y where X∼ exponential(0.5) and Y|X = x ∼ N(5,x2), find E(E(X|Y ))

A: Given, Two random variables X and Y such that X~Exp(0.5) and Y|X=x ~N(5,x2)

Q: If the moment generating function of a random variable X is: (1/3+(2/3)e t ) 5 find P (X > 3).

A: The given moment generating function of a random variable X is 13+23et5.

Q: Find fQ|X (q|x) for x ∈ {0, 1} and all q.

A: It is a fact of statistics. It is widely used.

Q: E(Y) –> P(Y > n).

A:

Q: Find the value of the constant c. Find P(YX). Find P(Y = X).

A: The probability mass function (PMF) of random variables X and Y is pX,Y(x,y)=c(x+y)2if x∈1,2,4 and…

Q: If X is a random variable, prove that Cov(X,X) = σX².

A: Let X is a random variable.

Q: 4. Let X be a random variable taking positive values and assume that E(X) exists. Which of the…

A: Option (I) is correct as E(X2)>= (E(X))2 when we find Variance.

Q: (1) E(E(X)) = E(X) (2) E(X – E(X)) = 0 (3) Var(X) = E(X²) – [E(X)]² (4) Given constants a and b,…

A: Discrete random variable is defined as a variable which can take only specific values. For example…

Q: (b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If…

A:

Q: X is a discrete random variable with p.m.f given by P(X=x)= x/c where c is a constant. The support…

A:

Q: ). Let M = ln(5*X) be a derived random variable. What is E[M]?

A:

Q: 4. If the random variable X has an show that for a > 0, b>0, exponential distrib f(x) = x/4, for x ≥…

A: For the random variable X~Exponential1λ, it is given that: fx=1λe-xλ, for x≥0

Q: Let X be a random variable with P(X = k) = 1/3 for k = −1, 0, 1. Let Y = X2 . (a) What is the p.m.f.…

A: Given X is a random variable with pmf P(X=k)  =  1/3 for k=-1,0,1 If Y= X2  Find  PMF of Y Are X…

Q: Suppose that X is a random variable for which E(X) = μ and Var(X) = σ2. Show that E[X(X − 1)] = μ(μ…

A:

Q: Let Z be a discrete random variable withE(Z)=0. Does it necessarily follow that E(Z3)=0? If yes,…

A: Solution

Q: Let X be a continuous random variable with pdf f(x) = x² for -1 < x < c. Find the value of c.

A: From the given information, the pdf of the random variable X is,

Q: Q2. Let X be a random variable with (a) Find E(X) (b) Find M(t). (0.42 f(x)=0.46 0.12 if x=0 if x =…

A: The random variable X f(x)=0.42         If x=0       0.46          If x=1      0.12          If…