# Find k if the following function represents the p. d. f. of a r. v. X. f(x) = {kx for 0<x<20 otherwise. Also find P[14<X<12] - Mathematics and Statistics

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

f(x) = {(kx,  "for"  0 < x < 2),(0,  "otherwise."):}

Also find "P"[1/4 < "X" < 1/2]

#### Solution

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

∴ int_0^2f(x)*dx = 1

∴ int_0^2"k"x*dx = 1

∴ "k" int_0^2 x*dx = 1

∴ "k"/(2)[x^2]_0^2 = 1

∴ "k"/(2)[4 - 0] = 1

∴ "k"/(2) = 1

∴ k = (1)/(2)

"P"[1/4 < "X" < 1/2] = int_(1/4)^(1/2)f(x)*dx

= int_(1/4)^(1/2)  x/(2)*dx

= (1)/(2) int_(1/4)^(1/2) x*dx

= (1)/(4)[x^2]_(1/4)^(1/2)

= (1)/(4)[1/4- 1/16]

= (1)/(4)[(4 - 1)/16]

= (3)/(64).

Concept: Probability Distribution of Discrete Random Variables
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Probability Distributions
Exercise 8.2 | Q 1.04 | Page 144