HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Important Questions

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)¹⁹ = 0⋅1348]

A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).

For the following probability density function (p. d. f) of X, find P(X < 1) and P(|x| < 1) 

`f(x) = x^2/18, -3 < x < 3`

            = 0,             otherwise

If X ∼ N (4,25), then find P(x ≤ 4)

The defects on a plywood sheet occur at random with an average of the defect per 50 sq. ft. What Is the probability that such sheet will have-

(a) No defects
(b) At least one defect 
[Use e^-1 = 0.3678]

A card is drawn at random and replaced four times from a well shuftled pack of 52 cards. Find the probability that -

(a) Two diamond cards are drawn.
(b) At least one diamond card is drawn.

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

Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.

A random variable X has the following probability distribution:

Determine:

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

Verify which of the following is p.d.f. of r.v. X:

 f(x) = sin x, for 0 ≤ x ≤ `π/2`

Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by f(x) = `1/5`, for 0 ≤ x ≤ 5 and = 0 otherwise.

Find the probability that waiting time is between 1 and 3.

Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by f(x) = `1/5`, for 0 ≤ x ≤ 5 and = 0 otherwise.

Find the probability that the waiting time is more than 4 minutes.

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).

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 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 k = ______