Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 7 Probability Distributions Ex 7.2 Questions and Answers.

## Maharashtra State Board 12th Maths Solutions Chapter 7 Probability Distributions Ex 7.2

Question 1.

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

(i) f(x) = sin x, for 0 ≤ x ≤ \(\frac{\pi}{2}\)

(ii) f(x) = x, for 0 ≤ x ≤ 1 and -2 – x for 1 < x < 2

(iii) fix) = 2, for 0 ≤ x ≤ 1.

Solution:

f(x) is the p.d.f. of r.v. X if

(a) f(x) ≥ 0 for all x ∈ R and

Hence, f(x) is the p.d.f. of X.

(ii) f(x) = x ≥ 0 if 0 ≤ x ≤ 1

For 1 < x < 2, -2 < -x < -1

-2 – 2 < -2 – x < -2 – 1

i.e. -4 < f(x) < -3 if 1 < x < 2

Hence, f(x) is not p.d.f. of X.

(iii) (a) f(x) = 2 ≥ 0 for 0 ≤ x ≤ 1

Hence, f(x) is not p.d.f. of X.

Question 2.

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

f(x) = \(\frac{x}{8}\), for 0 < x < 4 and = 0 otherwise.

Find

(a) P(x < 1.5)

(b) P(1 < x < 2) (c) P(x > 2).

Solution:

Question 3.

It is known that error in measurement of reaction temperature (in 0°C) in a certain experiment is continuous r.v. given by

f(x) = \(\frac{x^{2}}{3}\) for -1 < x < 2

= 0. otherwise.

(i) Verify whether f(x) is p.d.f. of r.v. X

(ii) Find P(0 < x ≤ 1)

(iii) Find the probability that X is negative.

Solution:

Question 4.

Find k if the following function represents p.d.f. of r.v. X

(i) f(x) = kx. for 0 < x < 2 and = 0 otherwise.

Also find P(\(\frac{1}{4}\) < x < \(\frac{3}{2}\)).

(ii) f(x) = kx(1 – x), for 0 < x < 1 and = 0 otherwise.

Also find P(\(\frac{1}{4}\) < x < \(\frac{1}{2}\)), P(x < \(\frac{1}{2}\)).

Solution:

(i) Since, the function f is p.d.f. of X

(ii) Since, the function f is the p.d.f. of X,

Question 5.

Let X be the amount of time for which a book is taken out of the library by a randomly selected students and suppose X has p.d.f.

f(x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.

Calculate:

(i) P(x ≤ 1)

(ii) P(0.5 ≤ x ≤ 1.5)

(iii) P(x ≥ 1.5).

Solution:

(i) P(x ≤ 1)

(ii) P(0.5 ≤ x ≤ 1.5)

(iii) P(x ≥ 1.5)

Question 6.

Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by f(x) = \(\frac{1}{5}\), for 0 ≤ x ≤ 5 and = 0 otherwise. Find the probability that

(i) waiting time is between 1 and 3

(ii) waiting time is more than 4 minutes.

Solution:

(i) Required probability = P(1 < X < 3)

(ii) Required probability = P(X > 4)

Question 7.

Suppose the error involved in making a certain measurement is a continuous r.v. X with p.d.f.

f(x) = k(4 – x^{2}), -2 ≤ x ≤ 2 and 0 otherwise.

Compute:

(i) P(X > 0)

(ii) P(-1 < X < 1)

(iii) P(-0.5 < X or X > 0.5).

Solution:

Since, f is the p.d.f. of X,

Question 8.

The following is the p.d.f. of continuous r.v. X

f(x) = \(\frac{x}{8}\), for 0 < x < 4 and = 0 otherwise.

(i) Find expression for c.d.f. of X.

(ii) Find F(x) at x = 0.5, 1.7 and 5.

Solution:

(i) Let F(x) be the c.d.f. of X

Question 9.

Given the p.d.f. of a continuous random r.v. X, f(x) = \(\frac{x^{2}}{3}\), for -1 < x < 2 and = 0 otherwise. Determine c.d.f. of X and hence find P(X < 1); P(X < -2), P(X > 0), P(1 < X < 2).

Solution:

Question 10.

If a r.v. X has p.d.f.

f(x) = \(\frac{c}{x}\) for 1 < x < 3, c > 0. Find c, E(X), Var (X).

Solution:

Since f(x) is p.d.f of r.v. X