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Chapter 3: Q1E (page 129)

Suppose that the joint p.d.f. of a pair of random variables (X,Y) is constant on the rectangle where 0鈮鈮�2 and 0鈮� y 鈮� 1, and suppose that the p.d.f. is 0 off of this rectangle.

a. Find the constant value of the p.d.f. on the rectangle.

b. Find Pr (X鈮)

Short Answer

Expert verified

a. The constant value is 0.50

b. The probability is 0.75

Step by step solution

01

Given information

The pdf of X is given by,


02

Finding the value of constant

03

Calculating the probability

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Most popular questions from this chapter

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\)form a random sample of nobservations from the uniform distribution on the interval(0, 1), and let Y denote the second largest of the observations.Determine the p.d.f. of Y.Hint: First, determine thec.d.f. G of Y by noting that

\(\begin{aligned}G\left( y \right) &= \Pr \left( {Y \le y} \right)\\ &= \Pr \left( {At\,\,least\,\,n - 1\,\,observations\,\, \le \,\,y} \right)\end{aligned}\)

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\) form a random sample of sizen from the uniform distribution on the interval [0, 1] andthat \({{\bf{Y}}_{\bf{n}}}{\bf{ = max}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}} \right)\). Find the smallest value of \({\bf{n}}\)such that\({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{n}}} \ge {\bf{0}}{\bf{.99}}} \right) \ge {\bf{0}}{\bf{.95}}\).

Question:Suppose that in a certain drug the concentration of aparticular chemical is a random variable with a continuousdistribution for which the p.d.f.gis as follows:

\({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{3}}}{{\bf{8}}}{{\bf{x}}^{\bf{2}}}\;{\bf{for}}\;{\bf{0}} \le {\bf{x}} \le {\bf{2}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

Suppose that the concentrationsXandYof the chemicalin two separate batches of the drug are independent randomvariables for each of which the p.d.f. isg. Determine

(a) the joint p.d.f.of X andY;

(b) Pr(X=Y);

(c) Pr(X >Y );

(d) Pr(X+Y鈮�1).

Suppose that thenrandom variablesX1. . . , Xnform arandom sample from a discrete distribution for which thep.f. is f. Determine the value of Pr(X1 = X2 = . . .= Xn).

Suppose that a Markov chain has four states 1, 2, 3, 4, and stationary transition probabilities as specified by the following transition matrix

\(p = \left[ {\begin{array}{*{20}{c}}{\frac{1}{4}}&{\frac{1}{4}}&0&{\frac{1}{2}}\\0&1&0&0\\{\frac{1}{2}}&0&{\frac{1}{2}}&0\\{\frac{1}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{1}{4}}\end{array}} \right]\):

a.If the chain is in state 3 at a given timen, what is the probability that it will be in state 2 at timen+2?

b.If the chain is in state 1 at a given timen, what is the probability it will be in state 3 at timen+3?
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