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Probability & Statistics Tutorial

The document outlines a tutorial for a Probability and Statistics course at the Birla Institute of Technology and Science, Pilani. It consists of various problems involving random variables, their distributions, and density functions, including transformations and independence of variables. The tutorial covers both discrete and continuous random variables with specific examples and calculations.

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13 views1 page

Probability & Statistics Tutorial

The document outlines a tutorial for a Probability and Statistics course at the Birla Institute of Technology and Science, Pilani. It consists of various problems involving random variables, their distributions, and density functions, including transformations and independence of variables. The tutorial covers both discrete and continuous random variables with specific examples and calculations.

Uploaded by

Deeksha
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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15

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI


K. K. BIRLA GOA CAMPUS
Second Semester 2019-2020
Tutorial-9

Course No. MATH F113 Course title: Probability and Statisitcs

(1) If X is uniformly distributed in (−1, 1), find g(x), so that the random variable Y = g(X) may
have the density function fY (y) = 2e−2y , y > 0.

(2) The random variable Y is defined by Y = 12 (X + |X|), where X is another random variable.
Determine the density and distribution function of Y in terms of those of X.

(3) Let Y = |X − 1| and fX (x) = 2e−2x , x > 0. Find fY (y).

(4) Let Y be a continuous random variable with density fX (x). Let Y = X 2 , find fY (y).

(5) Let X and Y be two independent random variables with identical probability density function
given by f (x) = e−x if x > 0. What is the probability density function of W = M ax{X, Y }?

(6) Let (X, Y ) be a two-dimensional random variable with PDF as follows


x\ y 1 2 3
2 1 3
1 18 18 18
3 2 1
2 18 18 18
1 3 2
3 18 18 18

Let Z = X + Y.
(i) Calculate mX (t), mY (t) and mZ (t).
(ii) Is mX (t)mY (t) = mZ (t)?
(iii) Are X and Y independent?

(7) Let (X, Y ) be of continuous type with joint density given by


1
fXY (x, y) = (1 + xy) |x| < 1, |y| < 1.
4
Are X and Y independent? Are X 2 and Y 2 independent? Justify.

(8) Let X and Y be independent and uniformly distributed random variables over (10, 20) and
(0, 10), respectively. Find the PDF of 12 (X − Y ).

(9) If the joint density of X and Y is given by


ye−x
fXY (x, y) = − ∞ < x < ∞, 0 < y < 2.
2(1 + e−x )2
1
Find the PDF of U = 1+e−X
.

(10) Let X and Y be iid RVs with common PDF


√ 2
f (x) = (x 2π)−1 e−(1/2)(ln(x)) x > 0.
Find PDF of Z = XY.

(11) Let X and Y be mutually independent with identical distribution. Suppose X and Y assume
only positive integral values and E(X) and E(Y ) exist. Let S = X + Y then calculate E( XS ).

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