1. The number of cars arriving at ICICI bank drive-in window during 10-min period is Poisson random variable X with b=2. What is the probability that more than 3 cars will arrive during any 10 min period?
a) 0.249
b) 0.143
c) 0.346
d) 0.543
Explanation: Evaluate 1 – P(x = 0) – P(x = 1) – P(x = 2) – P(x = 3).
2. The number of cars arriving at ICICI bank drive-in window during 10-min period is Poisson random variable X with b=2. What is the probability that no car will arrive is?
a) 0.516
b) 0.459
c) 0.246
d) 0.135
Explanation: Evaluate P(x = 0).
3. Delhi averages three murder per week and their occurrences follow a Poisson distribution. The probability that there will be five or more murders in a given week is?
a) 0.1847
b) 0.2461
c) 0.3927
d) 0.4167
Explanation: P(5 or more) = 1 – P(0) – P(1) – P(2) – P(3) – P(4) = 0.1847.
4. Delhi averages three murder per week and their occurrences follow a Poisson distribution. On average, how many weeks a year can Delhi expect to have no murders?
a) 1.4
b) 1.9
c) 2.6
d) 3.4
Explanation: P(0) = 0.0498. Hence average number of weeks per year with no murder is 52 x P(0) = 2.5889 week.
5. How many weeds per year (average) can the Delhi expect the number of murders per week to equal or exceed the average number per week?
a) 15
b) 20
c) 25
d) 30
Explanation: P(3 or more) = 1 – P(0) – P (1) – P(2) = 0.5768. Therefore average number of weeks per year = 52 x 0.5768 or 29.994 weeks.
6. The random variable X is defined by the density f(x) = 0.5u(x) e(0.5x). What is the expect value of g(x) = X3?
a) 48
b) 192
c) 36
d) 72
Explanation: Solve E[g(x)] = E[X3].
7. The random variable X is defined by the density f(x) = 0.5u(x) e(0.5x). What is the mean of the random variable x is?
a) 1/4
b) 1/6
c) 1/3
d) 1/5
Explanation: Solve integral (x f(x) dx) from negative infinity to x.
8. The random variable X is defined by the density f(x) = 0.5u(x) e(0.5x). What is the variance of the random variable x is?
a) 1/10
b) 3/80
c) 5/16
d) 3/16
Explanation: Variance is given by E[X230] – 1/16.
9. A joint sample space for two random variable X and Y has four elements (1,1), (2,2), (3,3) and (4,4). Probabilities of these elements are 0.1, 0.35, 0.05 and 0.5 respectively. What is the probability of the event{X 2.5, Y 6} is?
a) 0.45
b) 0.50
c) 0.55
d) 0.60
Explanation: The required answer is given by Fxy(2.5, 6.0) = 0.1 + 0.35 = 0.45.
10. A joint sample space for two random variable X and Y has four elements (1,1), (2,2), (3,3) and (4,4). Probabilities of these elements are 0.1, 0.35, 0.05 and 0.5 respectively. What is the probability of the event that X is less than three?
a) 0.45
b) 0.50
c) 0.55
d) 0.60
Explanation: The required answer is given by Fx(3.0) = Fxy(3.0, infinity) = 0.1 + 0.35 + 0.05 = 0.50.