Suppose that 20 of all copies of a particular textbook fail

Suppose that 20% of all copies of a particular textbook fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test. What is the probability distribution of X? E(X) and V(X) P(8 LE X LE 11)

Solution

total no. of copies selected = 15.

X = no. of copies that fail the test among these 15 copies!

define success as failing the test!

Prob. of succes = 0.2..

a) PDF of X = ( 15 C x) ( 0.2) ^ x ( 1 - 0.2) ^ ( 15 - x).....i.e, a binomial pdf!

b) E (X) = 15* 0.2 = 3...
VAR ( X) = 15* 0.2 * 0.8 = 2.4...

c) p [ 8 <= X < = 11 ] = p [ X =8]+ P[ X =9] + P [ X =10 ] + P [ X=11]

= P [ X <= 11 ] - P [ X <=7 ] = 0.999999 - 0.99576 = 0.004239...