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Sheet 6

This document contains 12 problems related to probability and joint probability distributions. The problems involve determining values that satisfy a joint distribution, finding probabilities and expectations from given joint distributions, and assessing independence and other properties of random variables described by joint distributions. Marginal distributions, conditional probabilities, covariance, and correlation are calculated for various pairs of random variables.

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113 views3 pages

Sheet 6

This document contains 12 problems related to probability and joint probability distributions. The problems involve determining values that satisfy a joint distribution, finding probabilities and expectations from given joint distributions, and assessing independence and other properties of random variables described by joint distributions. Marginal distributions, conditional probabilities, covariance, and correlation are calculated for various pairs of random variables.

Uploaded by

Akol Manyuop Lual
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Cairo University 1st Year Civil

Spring 2020
Faculty of Engineering Probability - Sheet 6

(1) Determine the values of c so that the following functions represent joint probability distributions
of the random variables X and Y:

(2) If the joint probability distribution of X and Y is given by

Find

(3) A fast-food restaurant operates both a drive through facility and a walk-in facility. On a
randomly selected day, let X and Y, respectively, be the proportions of the time that the drive-
through and walk-in facilities are in use, and suppose that the joint density function of these
random variables is

(a) Find the marginal density of X.


(b) Find the marginal density of Y .
(c) Find the probability that the drive-through facility is busy less than one-half of the time.
(d) Find the covariance of the random variables X and Y.
(4) Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature
(◦F) at which a certain reaction starts to take place. Suppose that two random variables X and
Y have the joint density

Find 𝑃(0 ≤ 𝑋 ≤ 0.5 and 0.25 ≤ 𝑌 ≤ 0.5)


(5) Each rear tire on an experimental airplane is supposed to be filled to a pressure of 40 pounds
per square inch (psi). Let X denote the actual air pressure for the right tire and Y denote the
actual air pressure for the left tire. Suppose that X and Y are random variables with the joint
density function

(a) Find k.
(b) Find P(30 ≤ X ≤ 40 and 40 ≤Y <50).
(c) Find the probability that both tires are underfilled.

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(6) Let X denote the number of times a certain numerical control machine will malfunction: 1, 2,
or 3 times on any given day. Let Y denote the number of times a technician is called on an
emergency call. Their joint probability distribution is given as

(a) Evaluate the marginal distribution of X.


(b) Evaluate the marginal distribution of Y.
(c) Find P(Y = 3 | X = 2).
(d) Find the covariance of the random variables X and Y.
(7) Given the joint density function

find P(1 < Y < 3 | X = 1).


(8) Consider the following joint probability density function of the random variables X and Y :

(a) Find the marginal density functions of X and Y.


(b) Are X and Y independent?
(c) Find P(X >2).
(9) Suppose that X and Y have the following joint probability function:

(10) Consider the joint density function


16𝑦
𝑓(𝑥, 𝑦) = { 𝑥 3 , 𝑥 ≥ 2, 0 ≤ 𝑦 ≤ 1
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Compute the correlation coefficient ρXY.

(11) Suppose that X and Y are independent random variables having the joint probability
distribution

Find
(a) E(2X − 3Y );
(b) E(XY ).
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(12) Let X and Y denote the lengths of life, in years, of two components in an electronic system.
If the joint density function of these variables is
𝑒 −(𝑥+𝑦) , 𝑥 > 0, 𝑦>0
𝑓(𝑥, 𝑦) = {
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
(a) Find 𝑃(0 < 𝑋 < 1 | 𝑌 = 2)
(b) Are X and Y independent?
(c) Find the covariance of the random variables X and Y.
(d) Compute the correlation coefficient ρXY.

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