Probability Homework with 4 problems

Assignment 1

Problem 1 (6 points)

a. (2 points) Do Problem 1.3, page 14 in the text. Explain.

b. (2 points) Let A, B, and C be events. Derive an expression for the probability P(A∪B ∪C). Your expression should generalize the expression

P(A∪B) = P(A) + P(B) −P(A∩B)

to unions of three events. Note that your expression should be in terms of the probabilities of the events A, B, and C and their intersections (not unions). Explain.

c. (2 points) Do Problem 1.23, page 15 in the text. Explain.

Problem 2 (14 points)

a. (4 points) Do Problem 2.9, page 80 in the text. Compute E[X], V ar(X), and the moment generating function φX(t), where t ∈ IR.

b. (7 points) Consider a random variable Y with probability density function (PDF):

fY (y) =

{ cye−y/5 if y > 0, 0 otherwise.

(i) Determine the value of c.

(ii) Completely specify the cumulative distribution function (CDF) of the random variable Y .

(iii) Use the probability density function (PDF) fY to compute E[Y ], V ar(Y ), and the moment generating function φY (t), where t ∈ IR.

(iv) Use the moment generating function φY to compute E[Y ] and V ar(Y ).

c. (3 points) Let Z be a normal random variable with mean 5 and variance 4 (i.e., Z ∼ N(5, 4)). (i) Use Table 2.3, page 76 in the text, to determine the probability

P{3.52 ≤ Z ≤ 9.14}. Explain. (ii) Define the random variable Z′ = 3Z − 7. Specify the distribution of Z′.

Explain.

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Problem 3 (15 points)

a. (2 points) Consider an exponential random variable X with rate λ > 0 (i.e., with probability density function (PDF) fX(x) = λe

−λx for all x ≥ 0 and fX(x) = 0 for all x < 0). Determine the conditional expected value E[X|X > 1/λ]. Explain.

b. (6 points) Do Problems 3.3 and 3.4, page 164 in the text. Compute the marginal probability mass function (PMF) and the marginal cumulative distribution func- tion (CDF) of the random variable Y .

c. (7 points) Consider random variables X and Y with joint probability density func- tion (PDF):

fX,Y (x,y) =

{ c if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, x + y ≥ 1, 0 otherwise.

(i) Determine the value of c.

(ii) Determine the probability P{0.5 ≤ X ≤ 1, 0 ≤ Y ≤ 0.5}. (iii) Compute the marginal probability density functions (PDFs) of the random

variables X and Y .

(iv) Compute E[Y ] and E[Y |X = x] for x = 2 3 .

(v) Are the random variables X and Y independent? Explain.

Problem 4 (10 points)

a. (6 points) Do Problem 3.40, page 170 in the text.

b. (4 points) Do Problem 3.26, page 167 in the text. Provide an intuition for your answer.

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