Introduction to Probability and Statistics for Engineers and Scientists

132 Chapter 4:Random Variables and Expectation

9. A bin of 5 transistors is known to contain 3 that are defective. The transistors are
   to be tested, one at a time, until the defective ones are identified. Denote byN 1
   the number of tests made until the first defective is spotted and byN 2 the number
   of additional tests until the second defective is spotted; find the joint probability
   mass function ofN 1 andN 2.
   10.The joint probability density function ofXandYis given by

f(x,y)=

6

7

(

x^2 +

xy

2

)

,0<x<1, 0<y< 2

(a) Verify that this is indeed a joint density function.

(b)Compute the density function ofX.

(c) FindP{X>Y}.

11.LetX 1 ,X 2 ,...,Xnbe independent random variables, each having a uniform distri-

butionover(0, 1). LetM=maximum(X 1 ,X 2 ,...,Xn). Showthatthedistribution

function ofM,FM(·), is given by

FM(x)=xn,0≤x≤ 1

What is the probability density function ofM?

12.The joint density ofXandYis given by

f(x,y)=

{

xe(−x+y) x>0,y> 0

0 otherwise

(a) Compute the density ofX.

(b)Compute the density ofY.

(c) AreXandYindependent?

13.The joint density ofXandYis

f(x,y)=

{

20 <x<y,0<y< 1

0 otherwise

(a) Compute the density ofX.

(b)Compute the density ofY.

(c) AreXandYindependent?

14.If the joint density function ofXandYfactors into one part depending only on

xand one depending only ony, show thatXandYare independent. That is, if

f(x,y)=k(x)l(y), −∞<x<∞, −∞<y<∞

show thatXandYare independent.