Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 11 Probability Distributions Ex 11.3 Textbook Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 11 Probability Distributions Ex 11.3

Question 1.

The probability density function of X is given by

Find the value of k.

Solution:

Since f(x) is a probability density function

Question 2.

The probability density function of X is

Find

(i) P(0.2 ≤ X < 0.6)

(ii) P(1.2 ≤ X < 1.8)

(iii) P(0.5 ≤ X < 1.5)

Solution:

Question 3.

Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function

Find

(i) the value of k

(ii) the distribution function

(iii) the probability that daily sales will fall between 300 litres and 500 litres?

Solution:

(i) Since f(x) is a probability density function

Question 4.

The probability density function of X is given by

Find

(i) the value of k

(ii) the distribution function

(iii) P(X < 3)

(iv) P(5 ≤ X)

(v) P(X ≤ 4)

Solution:

(i) Since f is a probability density function

(ii) The distribution function F(x) = \(\int_{-\infty}^{x}\) f(u) du

Question 5.

If X is the random variable with probability density function f(x) given by,

then find

(i) the distribution function F(x)

(ii) P(-0.5 ≤ x ≤ 0.5)

Solution:

Distribution function F(x) = \(\int_{-\infty}^{x}\) f(u) du

case 1.

x < -1

F(x) = \(\int_{-\infty}^{x}\) f(u) du = 0

Question 6.

If X is the random variable with distribution function F(x) given by,

Find

(i) the probability density function f(x)

(ii) P(0.3 ≤ X ≤ 0.6)

Solution:

(i) Probability density function

(ii) P(0.3 ≤ X ≤ 0.6)

P(a ≤ X ≤ b) = F(b) – F(a)

P(0.3 ≤ X ≤ 0.6) = F(0.6) – F(0.3)

= \(\frac { 1 }{ 2 }\) [(0.6)² + (0.6)] – \(\frac { 1 }{ 2 }\) [(0.3)² + (0.3)]

= \(\frac { 1 }{ 2 }\) (0.96 – 0.39)

= 0.285