Its your birthday and your friends are calling you up to congratulate you. The a

Question

Its your birthday and your friends are calling you up to congratulate you. The average waiting time, X, in minutes for the next call is an exponential random variable, X, with expected value 20 minutes and probability density function ex for x 0 and 0 otherwise.

a) What is the value of ? (X is measured in minutes)

b) What is the standard deviation of X?

c) What is cumulative distribution function of X for x 0

d) What is probability that you wait more than 10 minutes for the next call?

e) If you have already waited 10 minutes for a call then what is the probability you will have to wait at least 10 more minutes for thenext call?

f) What is the median of X?

g) If X1,X2,X3 are the next three calls then what is the probability all are less than 20 minutes

h) What theorem permits us to treat the sum or the sample mean of more than 30 independent, identically distributed random variables as if they were normal random variables?

i) What is the approximate probability that the total waiting time for the next 40 calls is between 790 and 810 minutes? That is, if X1,X2, ... ,X40 are waiting times for 40 calls then what is the approximate probability that 790 < X1 + X2 + ... +X40 < 810?

Explanation / Answer

SInce this is an exponential distribution, we use the formulas to calculate the first 4 parts.

1 ) = 1/Mean(X) = 1/20 min^-1

2) Standard deviation = mean = 20min

3) CDF = 1 ex=1 ex/20

4) Pr(X10)=e10/20=0.6065

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