EE3025 - Statistical Methods in ECE Spring 2024 Homework 5

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EE3025 - Statistical Methods in ECE

Spring 2024

Homework 5

Due on: Tuesday, March 19, 10:00 PM

Problem 1

You are offered to play the following game. You roll a fair die once and observe the result which is shown by the random variable X. At this point, you can stop the game and win X dollars. You can also choose to roll the die for the second time to observe the value Y . In this case, you will win Y dollars. Let W be the value that you win in this game. What strategy do you use to maximize E[W]? What is the maximum E[W] you can achieve using your strategy?

Hint: A natural strategy to choose is the folloing: if X > α then, stop the game after the first roll, otherwise go for the second roll. What should be α?

Problem 2

Let the cdf of a continuous random variable X is given by

(a) Draw this function, and justify that it is a valid cdf.

(b) what is the pdf of X? sketch it.

(c) What is P[X ≤ 0]? How about P[X > 4/1]?

(d) Compute P[−1/2 < X ≤ 1/2].

Problem 3

Let X ∼ N (0, σ2 ) be a Gaussian random variable.

(a) What are E[X] and E[X2 ]? You do not need to compute them. Rather you can just find them from distribution parameters.

(b) Compute E[|X|].

Hint: Use a change of variable y = 2σ2/x2.

(c) Compute E[X3] and E[X5]. What can you conclude?

(d) Compute E[X4].

Hint: Apply a change of variable y = x2, and use integration by parts.

(e) Let σ = 2. Compute P[X < 5] and P[−1 ≤ X ≤ 2].

Hint: You may use tables for Φ function.

Problem 4

Let Z be an exponential random variable with Var[Z] = 25.

(a) What is the pdf of Z?

(b) What is the second moment of Z?

(c) What is the probability that Z is greater than it’s first moment?

(d) Let Y = 2Z. Find P[Y ≤ k] as a function of k, and from which find the p.d.f. of Y .

Problem 5

Consider the generalized density

(a) Compute P(0 < X ≤ 7).

(b) Compute P(X = 0).

(c) Compute E[X2].