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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Sum

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]`

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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)[4]` = 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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Chapter 8: Probability Distributions - Exercise 8.2 [Page 144]

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

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