SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 2.7 - Probability Distributions [Latest edition]

Let the p.m.f. of a random variable X be P(x) = `(3 - x)/10`, for x = −1, 0, 1, 2 = 0, otherwise Then E(x) is ______

c.d.f. of a discrete random variable X is

If X denotes the number on the uppermost face of cubic die when it is tossed, then E(X) is ______

A random variable X has the following probability distribution

Then the variance of this distribution is

For the random variable X, if V(X) = 4, E(X) = 3, then E(x²) is ______

If a d.r.v. X takes values 0, 1, 2, 3, … with probability P(X = x) = k(x + 1) × 5^–x, where k is a constant, then P(X = 0) = ______

The p.m.f. of a d.r.v. X is P(X = x) = `{{:(((5),(x))/2^5",", "for"  x = 0","  1","  2","  3","  4","  5),(0",", "otherwise"):}` If a = P(X ≤ 2) and b = P(X ≥ 3), then

If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(x/("n"("n" + 1))",", "for"  x = 1","  2","  3","  .... "," "n"),(0",", "otherwise"):}`, then E(X) = ______

If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(("c")/x^3",", "for"  x = 1","  2","  3","),(0",", "otherwise"):}` then E(X) = ______

If a d.r.v. X has the following probability distribution:

then P(X = –1) is ______

If a d.r.v. X has the following probability distribution:

then k = ______

Let X represent the difference between the number of heads and the number of tails when a coin is tossed 6 times. What are the possible values of X?

An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X?

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

Find mean for the following probability distribution.

State if the following is not the probability mass function of a random variable. Give reasons for your answer

Find the expected value and variance of r.v. X whose p.m.f. is given below.

Find the probability distribution of number of heads in two tosses of a coin.

The probability distribution of X is as follows:

Find k and P[X < 2]

Solve the following problem :

Following is the probability distribution of a r.v.X.

Find the probability that X is positive.

Solve the following problem:

Following is the probability distribution of a r.v.X.

Find the probability that X is odd.

In the p.m.f. of r.v. X

Find a and obtain c.d.f. of X. 

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as number greater than 4 appears on at least one die.

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.

A random variable X has the following probability distribution:

Determine:

1. k
2. P(X < 3)
3. P( X > 4)

Find the probability distribution of the number of doublets in three throws of a pair of dice

Find the mean and variance of the number randomly selected from 1 to 15

Solve the following problem :

Let the p. m. f. of the r. v. X be

`"P"(x) = {((3 - x)/(10)", ","for"  x = -1", "0", "1", "2.),(0,"otherwise".):}`
Calculate E(X) and Var(X).

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as six appears on at least one die

Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X and variance of X

Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings drawn.

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).

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).

The following is the c.d.f. of r.v. X:

Find p.m.f. of X.
i. P(–1 ≤ X ≤ 2)
ii. P(X ≤ 3 / X > 0).

Solve the following problem :

A player tosses two coins. He wins ₹ 10 if 2 heads appear, ₹ 5 if 1 head appears, and ₹ 2 if no head appears. Find the expected value and variance of winning amount.

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.

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the standard deviation of X.