STA 4442/5440 Homework 5

Homework 5

STA 4442/5440

Problem 1: Consider two random variables X and Y with the following joint probability

mass function:

Y 0 1 4

X 1 0.2 0.05 0.25 3 0.05 0.2 0.25 5 0.15 0 0.05

(a) Find ρ(X,Y );

(b) Find V ar(X + Y );

(c) Find E(2X − √ Y ).

Problem 2: If a random variable X has the moment generating function

MX (t) = pet

1 − (1 −p)et ,

what is E(X)? Problem 3: It is known that, if X ∼ Poisson(λ), then its moment generating function is

MX (t) = exp(λ(e t − 1))

Now suppose X1,X2 are two independent Poisson random variables with pa- rameters λ1,λ2, respectively, what is the distribution of Y = X1 + X2?

Problem 4: Suppose X1, . . . ,X36 are independent standard normal random variables, what

is P(X̄ > 1)?

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