Probability and Statistics Questions with Solutions

Question 1
Part a
Using the formula given and substituting in for different values of beta and t; we are able to see
that the number of critical failures is the same across the time interval (5, 15) for beta equivalent
to 1. The expected number of critical failures decrease as beta tends to 0 and time increases from
5 to 15. Also, the expected number of critical failures increase for beta increase for B>1, as time
increases from 5 to 15.
Part b
To get the expected number critical failure we find the sum of the intensity function for beta=0.5
and time=5, 6, 7,..., 15. Hence:
E(t)=∑t=6
15
(0.5 βt(β−1)),forβ=0.5
E(t)=(0.5 β5(β−1))+(0.5β6(β−1))+...+(0.5 β15(β−1)),forβ=0.5
E(t)=0.91
The probability is given by
Exactly 2 hence x=2
Interval (5, 15)
Lambda= 0.5
p(x;λ)=¿ λxe−λ
x!
p(2;0.5)=¿ 0.52 e−0.5
2!
p(2;0.5)=¿0.075816
Part C

The cumulative distribution function is given by
F(x;λ)=∑x=0
∞λxe−λ
x!
And the probability density function
p(x;λ)=¿ λxe−λ
x!
They both have elements of the exponential distribution and it's important to note that Poisson
distribution is a subset of binomial distribution.
Question 2
Part a
(i)
Birth takes place when the process moves from state t to t+1 (the customers enter the store and
join the line) and dead takes place when the process moves from state t to t-1(the customer exists
the store because he finds a given number of people in the line). The process is subject to birth
rate (rate of customer entry into store), and death rate (rate of customer leaving).
(ii)
The customers arrive at rate 20 per hour hence 1 person per 3 minutes
The service time per customer is 3 minutes
Hence, the probability that the first customer would have left the queuing system before the
second customer arrives is one or 100%.
Part b
(i)