# If a r.v. X has p.d.f f(x) = {cx, 1<x<3,c>00, otherwise Find c, E(X), and Var(X). Also Find F(x). - Mathematics and Statistics

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If a r.v. X has p.d.f f(x) = {("c"/x","  1 < x < 3"," "c" > 0),(0","  "otherwise"):}
Find c, E(X), and Var(X). Also Find F(x).

#### Solution

a. Given that f(x) represents p.d.f. of r.v. X

∴ int_1^3 f(x)*"d"x = 1

∴ int_1^3 "c"/x*"d"x = 1

∴ "c" int_1^3 (1)/x*"d"x = 1

∴ "c"[logx]_1^3 = 1

∴ c [log 3 – log 1] = 1

∴ c [log 3 – 0] = 1

∴ c = (1)/log3

b. E(X) = int_(-oo)^(oo) xf(x)

= int_1^3 xf(x)*"d"x

= int_1^3 x "c"/x*"d"x

= "c" int_1^3 1*"d"x

= (1)/log3 [x]_1^3

= (1)/log3[3 - 1]

= (2)/log3.

c. E(X2) = int_(-oo)^(oo) x^2f(x)

= int_1^3 x^2f(x)*"d"x

= -int_1^3 x^2. "c"/x*"d"x

= "c" int_1^3x*"d"x

= (1)/(2log3)[x^2]_1^3

= (1)/(2log3) [9 - 1]

= 8/(2log3)

= (4)/(log3)

∴ Var(X) = E(X2) – [E(x)]2

= (4)/log3 -(2/log3)^2

= (4)/((log3)) -  4/(log3)^2

= (4log3 - 4)/(log3)^2

= (4(log3 - 1))/(log3)^2

F(x) = int_1^x f(x)*"d"x

= int_1^x "c"/x*"d"x

= "c" int_1^x (1)/x*"d"x

= "c"[logx]_1^x

= c[log x – log 1]

= c log x

= log x/log 3

Concept: Probability Distribution of a Continuous Random Variable
Is there an error in this question or solution?
Chapter 8: Probability Distributions - Exercise 8.2 [Page 145]

#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Probability Distributions
Exercise 8.2 | Q 1.1 | Page 145
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