Suppose the time spent by a randomly selected student who us

Suppose the time spent by a randomly selected student who use a terminal connected to a local time-sharing computer facility has a Gamma distribution with mean 20 min and variance 80 min².

B.What is the probability that a student uses the terminal for at most 24 min?

C.What is the probability that a student spends between 20 and 40 min using the terminal?

Solution

Given computer facility has a gamma distribution with mean 20 min and variance 80 min^2.

Our random variable for this problem is
=>X = minutes connected to timeshare computer

We are given that
=>X~Gamma(a, b )

But we do not know the values of a and b .

We are asked to find:
P(X 24) and P(20<X<40)

a) To Compute a and b
We know , the parameters a and b can be found using the relationship between the
parameters of the distribution and E(X) and V(X). we saw that for
the Gamma distribution
=>b = V(X)/E(X)
and
=>a = E(X)/b

For our problem these equations give us a = 5 and b = 4.

b) One way to calculate this probability is to use Excel.
P(X< 24) = F(24)
= 0.7149

A second way to calculate this probability is to use the standard Gamma
Chart.
=>P(X< 24) = P(G < 24/4) (standardize)
=> F(6;5)
=> 0.715

Hence the probability that a student uses the terminal for at most 24 min =0.715

c)One way to calculate

This probability is to use Excel.
=>P(20 <X < 40) = P(X<40) - P(X<20)
=>F(40) - F(20)
=>0.4112

F(40) - F(20) can be found in Excel by using the GAMMADIST function in
the following way

A second way to calculate this probability is to use the standard Gamma
Chart.
=>P(20 <X < 40) = P(X<40) - P(X<20)
=>P(G < 40/4) - P(G < 20/4) (standardize)
=> F(10;5) - F(5;5)
=> 0.971 - 0.560 (found in the Gamma table)

=>0.411

Hence the probability that a student spends between 20 min and 40 min using the terminal=0.411