Problem Set 2.pdf

Problem Set 2 Probability and random variables HS550: Statistical Methods Feb-Jun 2019

1.

In a family with 4 children, what is the probability that there will be two boys and two girls?

2.

In the first game Méré wins if in 4 rolls of a fair die there is atleast one 6. In a second game he wins if in 24 rolls of two fair dice there is atleast one pair of 6s. In which game he has a higher probability of win?

3.

A bag contains 100 pens, 50 of them are red and the rest are blue. Select 5 pens at random at one go. What is the probability that 3 are blue and 2 are red? If you draw them one by one, what is the probability that they will appear in the order BBBRR?

4.

Three urns contain, respectively 4 white and 2 black chips, 2 white and 4 black chips, 3 white and three black chips. Suppose a chip is chosen at random and is found to be white. What is the probability that the first urn is chosen?

5.

What is the probability of drawing a king and a queen consecutively from a deck of 52 cards, without replacement?

6.

Let X be a discrete random variable assuming probability (i.e. the probability mass function is given as) Px (x) = ½ for x=0 1/3 for x=1 1/6 for x=2 0 otherwise Find the range of the random variable X. Find P(X≥1.5). Find Prob(0≤X≤2). What is E(X) and V(X)

7.

Let X be a continuous random variable with the following probability density function: FX (X) = ce-x for x≥0, 0 otherwise Where c is a positive constant. Find c. Find the cumulative distribution function of X. Find Prob(1≤X≤3)

8.

Let X be a continuous random variable with the following probability density function: FX (X) = 2x for 0 ≤x ≤1, 0 otherwise Find expected value of X

9.

Assuming that the probability of a new born to be a boy is 0.51, find the probability distribution of number of boys in a family of four children. In a sample of 500 couples how many do you expect to have a) exactly 1 boy, b) exactly 1 girl, c) at least 1 boy?

10. An unbiased coin is thrown three times. If the random variable X denotes the number of heads obtained, find out the cumulative distribution function of X. 11. The probability density function of a random variable is given by FX (X) = k(x-1)(2-x) where 1 ≤x ≤2 Determine the following: a. The value of the constant k b. The cumulative distribution functions c. The probability that the random variable lies between 5/4 and 3/2