Department of Mathematics, IIT MadrasMA 2040 : Probability, Statistics and Stochastic Processes

January - May 2015

Problem Set - IV

27.02.2015

1. Let X be a random variable with PDF fX . Find the PDF of the random variable Y = |X|

(a) when

fX(x) =

{1/3, if − 2 < x ≤ 1,

0, otherwise;

(b) when

fX(x) =

{2e−2x, if x > 0,

0, otherwise;

(c) for general fX(x).

2. An ambulance travels back and forth, at a constant specific speed v, along a road of length l. We

may model the location of the ambulance at any moment in time to be uniformly distributed over the

interval (0, l). Also at any moment in time, an accident (not involving the ambulance itself) occurs at

a point uniformly distributed on the road; that is, the accidents distance from one of the fixed ends

of the road is also uniformly distributed over the interval (0, l). Assume the location of the accident

and the location of the ambulance are independent. Supposing the ambulance is capable of immediate

U-turns, compute the CDF and PDF of the ambulances travel time T to the location of the accident.

3. Let X be a discrete random variable with PMF pX and let Y be a continuous random variable,

independent from X, with PDF fY . Derive a formula for the PDF of the random variable X + Y .

4. The random variables X and Y are described by a joint PDF which is constant within the unit area

quadrilateral with vertices (0, 0), (0, 1), (1, 2), and (1, 1). Use the law of total variance to find the

variance of X + Y .

5. (a) You roll a fair six-sided die, and then you flip a fair coin the number of times shown by the die.

Find the expected value and the variance of the number of heads obtained.

(b) Repeat part (a) for the case where you roll two dice, instead of one.

6. The transform of the distribution of a random variable X (also referred to as the moment generating

function of X) is a function MX(s) of a free parameter s, defined by

MX(s) = E[esX ].

Compute the moment generating functions for the following distributions

(a) For X, having binomial distribution with parameters n and p, where n is positive integer and

0 < p < 1.

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(b) For Y , having Poisson distribution with parameter λ > 0.

(c) For Z, having geometric distribution with parameter p, where 0 < p < 1.

(d) For U , having exponential distribution with parameter λ > 0.

(e) For V , having normal distribution with parameters µ and σ, where −∞ < µ <∞ and σ > 0.

7. Let A = XY and L = 2(X + Y ) denote the area and the perimeter of a rectangle, respectively, where

X and Y denote the length and breadth of the rectangle. Assume that X and Y are independent, and

uniformly distributed on the interval [0, 1]:

(a) Find E[A] and E[L]

(b) Find Var(A)

(c) Find Var(L)

(d) Find Cov(A,L)

(e) Find the correlation coefficient ρ(A,L).

(f) Check that ρ(A,L) is less than, but fairly close to, one. Why?

8. The continuous random variables X and Y have joint pdf

fX,Y (x, y) =

{e−x−y, if 0 < x <∞, 0 < y <∞,0, otherwise.

Determine the pdf of the random variable Z = X + Y.

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