Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3

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
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 1
Find the value of k.
Solution:
Since f(x) is a probability density function
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 2

Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3

Question 2.
The probability density function of X is
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 3
Find
(i) P(0.2 ≤ X < 0.6)
(ii) P(1.2 ≤ X < 1.8)
(iii) P(0.5 ≤ X < 1.5)
Solution:
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 4
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 5

Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3

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
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 6
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:
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 21
(i) Since f(x) is a probability density function
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 7
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 8
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 9

Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3

Question 4.
The probability density function of X is given by
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 10
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
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 11

Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3

(ii) The distribution function F(x) = \(\int_{-\infty}^{x}\) f(u) du
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 12
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 13

Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3

Question 5.
If X is the random variable with probability density function f(x) given by,
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 14
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
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 15
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 16
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 17
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 18

Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3

Question 6.
If X is the random variable with distribution function F(x) given by,
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 19
Find
(i) the probability density function f(x)
(ii) P(0.3 ≤ X ≤ 0.6)
Solution:
(i) Probability density function
Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3 20

(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

Samacheer Kalvi 12th Maths Guide Chapter 11 Probability Distributions Ex 11.3

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