Students must practice these TS Inter 2nd Year Maths 2A Important Questions Chapter 10 Random Variables and Probability Distributions to help strengthen their preparations for exams.

## TS Inter 2nd Year Maths 2A Random Variables and Probability Distributions Important Questions

Question 1.

A cubical die is thrown. Find the mean and variance of X, giving the number on the face that shows up.

Solution:

Let S be the sample space and X be the random variable associated with S, where P(X) is given by the following table.

Question 2.

The probability distribution of a random variable X is given below:

Find the value of k and the mean and variance of X.

Solution:

Question 3.

If x is a random variable with probability distribution p(X=k)=\(\frac{(k+1) c}{2^k}\) k = 0,1,2 ………… then find c.

Solution:

Since p(X=k)=\(\frac{(k+1) c}{2^k}\) k = 0,1,2 ……….. is the probability distribution of x

Question 4.

Let X be a random variable such that

P(x=-2) = P(X = -1) = P(X=2)

P(X = 1) = \(\frac{1}{6}\) and P(X = 0) \(\frac{1}{3}\)

Find the mean and variance of X.

Solution:

Question 5.

Two dice are roiled at random. Find the probability distribution of the sum of the numbers on them. Find the mean of the random variable.

Solution:

When two dice are rolled, the sample space S consists of 6 x 6 = 36 sample points

S = ((1, 1), (1, 2), (1, 6), (2, 1), (2, 6),(6, 6)).

Let X denote the sum of the numbers on the two dice. Then the range of X = {(2, 3, 4 , 12)}

The probability distribution for X is given here under:

Question 6.

8 coins are tossed simultaneously. Find the probability of getting atleast 6 heads.

Solution:

In the experiment of tossing a coin, the probability of getting a head \(\frac{1}{2}\) and the probability of getting a tail \(\frac{1}{2}\). The probability of getting r heads in a random throw of 8 coins is

Question 7.

The mean and variance of a binomial distribution are 4 and 3 respectively. Fix the distribution and find P(X≥1).

Solution:

Here x = B(n,p) is specified by np = 4 = μ and npq = σ^{2} = 3

Question 8.

The probability that a person chosen at random is left handed (in hand wilting) is 0.1. What is the probability that in a group of 10 people, there is one who is left handed.

Solution:

Heye n = 10, find p =\(\frac{1}{10}\) = 0.1.

Hence q = 0.9

We have to find P(X = 1); the probability that

exactly one out of 10 is left handed

P(X = 1) = ^{10}C_{1} p^{1} q^{10-1}

= 10 x 0.1 x (0.9)^{9} = (0.9)^{9}

Question 9.

In a book of 450 pages, there are 400 typo graphical errors. Assuming that the number of errors per page follow the Poisson law, find the probability that a random sample of 5 pages will contain no typo graphical error.

Solution:

The average number of errors per page in the book is \(\lambda=\frac{400}{450}=\frac{8}{9}\)

The probability that there are r errors per page:

Hence P(X=0) = e^{-8/9}

The required probability that a random sample of 5 pages will contain no error is [P(X) = 0)]^{5} = (e^{-8/9})^{5}

Question 10.

The deficiency of red cells In the blood cells is determined by examining a specimen of blood under a microscope. Suppose a small fixed volume contains on an average 20 red cells for normal persons. Using the Poisson distribution, find the probability that a specimen of blood taken from a normal person will contain less than 15 red cells.

Solution:

Here λ = 20.

Let P(X = r) denote the probability that a specimen taken from a normal person will contain r red cells.

Then we have P(X < 15)

Question 11.

A Poisson variable satisfies P(X = 1) = P(X = 2). Find P(X = 5).

Solution: