Calculus and Above

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I am preparing for a...

I am preparing for a probability theory test and I am having problems with MGF's these are some examples from previous exams which I would like solutions for along with explanations to help me study. There is 5 Q's in total.

10. a)Y has moment generating function mY (z) = e^−8z+32z^2 . What is the

variance of Y ? What is P(Y ≤ −4)?

b)Use the moment generating function to derive the expected value of a

Poisson random variable with parameter λ = 1/3.

c)Let Y ∼ Uniform(α,β), α,β ∈ R and α β−Y}= 1/(1+y)

9. (a)

The random variable X has moment generating function

mX (t) = exp (−5t + 12t^2) .

If Z = 1/5 (X − 2), what are E(Z) and Var(Z)?

￼(b) Y is a random variable with moment generating function mY (t) = (1/(1 − 2t))^8 .

What are E(Y ) and Var(Y )?

8.

Let Z1 and Z2 be independent normal random variables where Z1 ∼ N(6,3^2) and Z2 ∼ N(4,1^2). Define Y1 = Z1 −Z2 and Y2 = 2(Z1) +3(Z2). Find the means and variances of the random variables Y1 and Y2. [6]

The random variable X has moment generating function mX (t) = exp(−7t + 18t^2).

If W = 1/5(X − 2) what are the expected value and variance of W?

10. (a)

The random variable X has moment generating function

mx(t)=(1−2t)^−15/2 .

If Z = 1/3(X + 3), what is the variance V(Z)?

￼(b) Y is a random variable with moment generating function mY (t) = (0.8 + 0.2e^t)^10 .

What is P(1≤Y ≤4)?

11. (a)

The random variable X takes the values 0, 1, 2 with probabilities 1/2 , 3/8, 1/8

respectively. Find the moment generating function for X and verify that

￼￼￼the second moment of X is 7/8.

[5]

(b)

The random variable Z has distribution function FZ (z) = 1 − exp(−5z) for z ∈ [0, ∞). Find the moment generating function for Z and verify

that the first moment of Z is 0.2.

[5]

10. a)Y has moment generating function mY (z) = e^−8z+32z^2 . What is the

variance of Y ? What is P(Y ≤ −4)?

b)Use the moment generating function to derive the expected value of a

Poisson random variable with parameter λ = 1/3.

c)Let Y ∼ Uniform(α,β), α,β ∈ R and α β−Y}= 1/(1+y)

9. (a)

The random variable X has moment generating function

mX (t) = exp (−5t + 12t^2) .

If Z = 1/5 (X − 2), what are E(Z) and Var(Z)?

￼(b) Y is a random variable with moment generating function mY (t) = (1/(1 − 2t))^8 .

What are E(Y ) and Var(Y )?

8.

Let Z1 and Z2 be independent normal random variables where Z1 ∼ N(6,3^2) and Z2 ∼ N(4,1^2). Define Y1 = Z1 −Z2 and Y2 = 2(Z1) +3(Z2). Find the means and variances of the random variables Y1 and Y2. [6]

The random variable X has moment generating function mX (t) = exp(−7t + 18t^2).

If W = 1/5(X − 2) what are the expected value and variance of W?

10. (a)

The random variable X has moment generating function

mx(t)=(1−2t)^−15/2 .

If Z = 1/3(X + 3), what is the variance V(Z)?

￼(b) Y is a random variable with moment generating function mY (t) = (0.8 + 0.2e^t)^10 .

What is P(1≤Y ≤4)?

11. (a)

The random variable X takes the values 0, 1, 2 with probabilities 1/2 , 3/8, 1/8

respectively. Find the moment generating function for X and verify that

￼￼￼the second moment of X is 7/8.

[5]

(b)

The random variable Z has distribution function FZ (z) = 1 − exp(−5z) for z ∈ [0, ∞). Find the moment generating function for Z and verify

that the first moment of Z is 0.2.

[5]

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