553.420/620 Probability

Assignment #08

1. Consider the jointly continuous rvs X and Y with joint pdf f(x, y) = xe−x(1+y) for x > 0 and y > 0; f(x, y) = 0 elsewhere.

(a) Compute P(Y ≤ 1|X ≤ 1).

(b) Compute P(Y ≤ 1|X = 1).

2. A machine makes random rectangles. The length (X) and width (Y ) are independent random variables: X ∼ uniform(0, 1), Y ∼ uniform(0, 1).

(a) What proportion of rectangles have area greater than 1/2?

(b) Derive the pdf of A = XY , i.e., the area of a rectangle. Use the CDF method.

(c)** What proportion of rectangles having area = 1/2 have length greater than 3/4?

**For part (c), you will need to find the conditional pdf of X|A, this requires the joint pdf of X and A. This may require you use the method of Jacobians (take U = X, V = XY =⇒ X = U, Y = V /U).

3. Use the method of convolutions to show that the sum of two independent geom(p) rvs follows a neg.binom(2, p) distribution.

4. Suppose X ∼ uniform(0, 1) and Y ∼ uniform(0, 2) are independent. Use the method of convolutions to construct the PDF of X + Y .

5. (a sum of independent normals is normal) Let X1, X2, . . . , Xn be independent and, for each i, Xi ∼ N(µi , σi 2 ). Use the MGF method to show that ∑ni=1 Xi has a normal distribution and identify the mean and the variance of this normal distribution.

Remark 1. The result of this problem is very important to remember (especially for the sequel course).

Remark 2. You should think about what this problem says in the i.i.d. case, i.e., when all the normal Xi ’s have the same parameters, say µ, σ2 .

6. Let N be the number of customers that enter a facility. Suppose that N ∼ Poisson(λ). Let X be the number of customers that don’t buy anything while in the facility. Assume that given N = n, X ∼ binom(n, p).

(a) Derive the (unconditional) distribution of X and identify it if you can.

(b) Compute P(X = 0).

7. A person shows up to work X hours after 12:00PM (time 0), where X ∼ uniform(0, 4). If they arrive at time X, they work an amount of time Y that has an exponential distribution with parameter 5 − X. What’s the probability they are still working after time 8?

8. Suppose Y |X = x ∼ Exp(x) and X ∼ Gamma(α, 1).

(a) Write down the PDF’s that are given to you in the problem statement.

(b) Write down the joint PDF of X, Y . (c) Derive the (marginal) distribution of Y . If this distribution has a name, name it. Be specific.

(d) Derive the conditional PDF of X given Y = y. If this distribution has a name, name it. Be specific.

9. Suppose Y ∼ uniform(0, X), where X follows the gamma density fX(x) = xe−x for x > 0.

(a) Derive the PDF of Y . Identify the distribution of Y .

(b) Let U = Y and V = X − Y . Construct the joint PDF of U and V using method of Jacobians.

10. Let X ∼ χ 2 m and Y = χ 2 n be independent. Let U = X/m Y /n = m n X Y . Derive the PDF of U. The distribution of U is called the F-distribution with m numerator degrees of freedom and n denominator degrees of freedom. Remark. To use the method of Jacobians you’ll need to choose a V that makes the transformation (x, y) 7→ (u, v) one-to-one. Although any such choice will lead to a correct marginal, I suggest V = Y because it might makes calculations more straightforward.

11. Suppose X and Y are random variables having the joint PDF fX,Y (x, y) = 4xy for 0 < x < 1, 0 < y < 1. Let U = X2 and V = XY . Derive the joint PDF of U, V .

12. X and Y have the joint pdf fX,Y (x, y) = e −y for 0 < x < y < ∞ (= 0 otherwise). Use the method of Jacobians to find the pdf of U = Y − X.