Question

5. Bob is supposed to take an exam. There are 5 questions and the event that each question is correctly answered is 30% and those events are independent. Let X be a random variable for the number of correct answers that Bob will make.

(a) Calculate P(X ≥ 1).

(b) Calculate P(X = 2).

(c) Obtain the cumulative distribution function for X.

(d) Calculate the mean and the variance of X.

6. Ann and Bob play a simple game: each independently draws a number from 1 to 10 each of which is equally likely. The person with a higher number wins. If they draw a same number, then there is a draw.
(a) What is the probability of Ann’s winning before drawing a number?

(b) What is the probability of Ann’s winning conditional on drawing number 7?

(c) Let X1 be a random variable for the number Ann draws; and let X2 be that for Bob. Calculate P(X1 + X2 ≥ 18).

(d) What is the mean of X1 + X2? What is the variance?

Answer 1

5:

Here X has binomial distribution with following parameters

n = 5 and p = 0.30

(a)

(b0

(c)

The CDF is

Following table shows the cumulative function of X:

X	P(X =x)	P(X <=x)
0	0.16807	0.16807
1	0.36015	0.52822
2	0.3087	0.83692
3	0.1323	0.96922
4	0.02835	0.99757
5	0.00243	1

(d)

The mean is

The variance is