1. State which of the following are not the probability distributions of a random variable. Give reasons for your answer. (i) X 0 1 2 P(X) 0.4 0.4 0.2 (ii) X 0 1 2 3 4 P(X) 0.1 0.5 0.2 0.1 0 (iii) Y 1 0 1 P(Y) 0.6 0.1 0.2 (iv) Z 3 2 1 0 1 P(Z) 0.3 0.2 0.4 0.1 0.05 2. An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable ? 3. Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X? 4. Find the probability distribution of (i) number of heads in two tosses of a coin. (ii) number of tails in the simultaneous tosses of three coins. (iii) numbe
Question 1:State which of the following are not the probability distributions of a random variable. Give reasons for your answer. (i) X 0 1 2 P(X) 0.4 0.4 0.2 (ii) X 0 1 2 3 4 P(X) 0.1 0.5 0.2 – 0.1 0 (iii) Y – 1 0 1 P(Y) 0.6 0.1 0.2 (iv) Z 3 2 1 0 –1 P(Z) 0.3 0.2 0.4 0.1 0.05
Question 2:An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable ?
Question 3:Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?
Question 4:Find the probability distribution of (i) number of heads in two tosses of a coin. (ii) number of tails in the simultaneous tosses of three coins. (iii) number of heads in four tosses of a coin.
Question 5:Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as (i) number greater than 4 (ii) six appears on at least one die
Question 6:From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
Question 7:A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.
Question 8:A random variable X has the following probability distribution: X 0123456 7 P(X) 0 k 2k 2k 3k k 2 2k 2 7k 2+k Determine (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(0 < X < 3)
Question 9:The random variable X has a probability distribution P(X) of the following form, where k is some number : P(X) = , 0 2, 1 3, 2 0, otherwise k if x k if x k if x ✂ (a) Determine the value of k. (b) Find P (X < 2), P (X 2), P(X 2).
Question 10:Find the mean number of heads in three tosses of a fair coin.
Question 11:Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Question 12:Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).
Question 13:Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
Question 14:A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.
Question 15:In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var (X). Choose the correct answer in each of the following:
Question 16:The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is (A) 1 (B) 2 (C) 5 (D) 8 3
Question 17:Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then the value of E(X) is (A) 37 221 (B) 5 13 (C) 1 13 (D) 2
