Exercise: Discrete Random Variable

1) Three balls are chosen from an urn containing 3 white, 3 red, 5 black balls. Suppose we win Rs. 10 for each white ball and lose Rs. 10 for each red ball selected. Let X denotes the total wining from this experiment, then what is the probability distribution of X? What amount do you expect to win?

2) It is known that screws produced by a certain company will be defective with probability 0.01 independently of each other. The company sells the screws in the package of 10 and offers a money-back guarantee that at most 1 of 10 screws is defective. What proportion of the packages sold must the company replace?

3) The probability mass function of a random variable X is given by p(i) = cλi/i! for i= 0,1,2,3,….., where λ is some positive value. Find (a) P[X=0] and P[X>2].

4) A school class of 120 students are driven in 3 buses to a sports ground. There are 36 students in one of the buses, 40 in another, and 44 in third bus. When the buses arrive, one of the 120 students is randomly chosen. Let X denote the number of students on the bus of that randomly chosen student, and find E[X].

5) Two balls are chosen randomly from an urn containing 8 white, 4 black and 2 orange balls. Suppose that we win $2 for each black ball selected and we lose $1 for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with each value?

6) Two fair dice rolled. Let X equal the product of the two numbers appear in two dice. Compute the probability mass function of X.

7) If the cumulative distribution function of X is given by,

F(b) = 0 if b < 0

=1/2 if 0 ≤ b <1

=3/5 if 1 ≤ b < 2

=4/5 if 2 ≤ b < 3

=9/10 if 3 ≤ b < 3.5

=1 if 3.5 ≤ b

Calculate the p.m.f of X.

8) A person tosses a fair coin until a tail appears for the first time. If the tail appears in the n-th flip, the person wins $2^n. Let X denote the player's winnings. Show that E(X) is not finite. This problem is known as the St. Petersburg paradox.

(a) Would you be willing to pay $ 1 million to play this game once?

(b) Would you be willing to pay $ 1 million for each game if you could for as long as you like and only had to settle up when you stopped playing?