Your stockbroker is free to answer your phone calls about 73

Your stockbroker is free to answer your phone calls about 73% of the time; otherwise he is talking to another client or is out of the office. You call him at five random times during a given month. We can assume that each call is independent of each other.

a) Let S be the number of times that your stockbroker answers your phone calls in the given month. What are the distribution, parameter(s) and support of S?

b) What is the probability that he will answer every one of the calls?

c) What is the probability that he will answer exactly four of your calls?

d) What is the probability that he will answer at least two of the calls?

e) Knowing that he answered at least two of your calls, what is the probability that he answered fewer than four of your calls?

f) What is the standard deviation of the number of phone call that you stockbroker will answer?

g) Each answered phone call lasts an average of 8.2 minutes. How much time do you expect to talk to your stockbroker in the given month?

Solution

a)
Binomial Distribution

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

b)
P( X = 5 ) = ( 5 5 ) * ( 0.73^5) * ( 1 - 0.73 )^0
= 0.2073
c)
P( X = 4 ) = ( 5 4 ) * ( 0.73^4) * ( 1 - 0.73 )^1
= 0.3834
d)

P( X < 2) = P(X=1) + P(X=0)
= ( 5 1 ) * 0.73^1 * ( 1- 0.73 ) ^4 + ( 5 0 ) * 0.73^0 * ( 1- 0.73 ) ^5
= 0.0208
P( X > = 2 ) = 1 - P( X < 2) = 0.9792

Note: Post the remainig parts in next post