2 Suppose that 30 of all drivers stop at an intersection hav

2. Suppose that 30% of all drivers stop at an intersection having flashing red lights when no other cars are visible. Of 15 randomly selected drivers coming to an intersection under these conditions, let X denote the number of those who stop. (a) The random variable X is (choose one) (i) binomial (ii) hypergeometric (iii) negative bino- (i) binomial) hypergeometric (i) negative bino- mial (iv) Poisson.

Solution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
a)
Binomial
b)
Normal Approximation to Binomial Distribution
Mean ( np ) =15 * 0.3 = 4.5
Standard Deviation ( npq )= 15*0.3*0.7 = 1.7748

c)
P( X = 6 ) = ( 15 6 ) * ( 0.3^6) * ( 1 - 0.3 )^9
= 0.1472

P( X < 6) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 15 5 ) * 0.3^5 * ( 1- 0.3 ) ^10 + ( 15 4 ) * 0.3^4 * ( 1- 0.3 ) ^11 + ( 15 3 ) * 0.3^3 * ( 1- 0.3 ) ^12 + ( 15 2 ) * 0.3^2 * ( 1- 0.3 ) ^13 + ( 15 1 ) * 0.3^1 * ( 1- 0.3 ) ^14 + ( 15 0 ) * 0.3^0 * ( 1- 0.3 ) ^15
= 0.7216
P( X > = 6 ) = 1 - P( X < 6) = 0.2784