Answer to Question #154558 in Statistics and Probability for 75ed86

Question

A box contains ten light bulbs of which four are defective. A bulb is selected from the box and tested. If it is defective, another bulb is selected and tested, until a non-defective bulb is chosen. Let the random variable X be the number of bulbs chosen. 

a) Write sample space for this random experiment?

b) Find probability mass Function of X? 

c) Find CDF of X?

Answer 1

a) Write sample space for this random experiment?

RX = {1,2,3,4,5,6,7}

b) Find probability mass Function of X? 

$P(X=1) = \frac4{10}=0.4\\ P(X=2) = \frac6{10}\frac49 = 4/15=0.2667\\ P(X=3) = \frac6{10}\frac59\frac48=1/6=0.1667\\ P(X=4) = \frac6{10}\frac59\frac48\frac47=2/21=0.0952\\ P(X=5) = \frac6{10}\frac59\frac48\frac37\frac46=1/21=0.0476\\ P(X=6) = \frac6{10}\frac59\frac48\frac37\frac26\frac45=2/105=0.0190\\ P(X=7) = \frac6{10}\frac59\frac48\frac37\frac26\frac15\frac44=1/210=0.048$

c) Find CDF of X?

$P(X\le1) =0.4\\ P(X\le2) =2/5+4/15=2/3=0.6667\\ P(X\le3) = 2/3+1/6=5/6=0.8333\\ P(X\le4) = 5/6+2/21=13/14=0.9286\\ P(X\le5) = 13/14+1/21=41/42=0.9762\\ P(X\le6) = 20/21+2/105=209/210=0.9952\\ P(X\le7) = 1.0$