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The probability density function of the random variable X is given by ef(x)={16xe-4xx>00x≤0Find the mean and variance of X

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प्रश्न

The probability density function of the random variable X is given by

`f(x) = {{:(16x"e"^(-4x), x > 0),(0, x ≤ 0):}`
find the mean and variance of X

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उत्तर

Given p.d.f is `f(x) = {{:(16x"e"^(-4x), x > 0),(0, x ≤ 0):}`

Mean: E(X)

= `int_-oo^oo x f(x) "d"x`

= `16int_0^oo x^2 "e"^(-4x) "d"x`

 Using integration by parts method twice

Let u = x2

⇒ du = 2x dx

And `int "dv" = int "e"^(-4x)`

v = `"e"^(-4x)/(-4)`

`int "u" "d" = "uv" - int "v"  "du"`

`int^2"e"^(-4x) "d"x = (x^2"e"^(-4x))/(-4) + 1/4 int 2x"e"^(-4x) "d"x`  .......(1)

= `- (x^2"e"^(-4x))/4 + 1/2 int x"e"^(-4x) "d"x`

∵ Integration by parts method

u = x

⇒ du = dx

And `int "dv" - int"e"^(-4x) ""x`

v = `"e"^(-4x)/(-4)`

`int "u"  "dv" = "uv" - int "v"  "du"`

`int x"e"^(-4x) "d"x = (-x"e"^(-4x))/4 + 1/4 int "e"^(-4x) "d"x`

= `(-x"e"^(-4x))/4 - 1/16 "e"^(-4x)`

Substituting in (1)

`intx^2"e"^(-4x) "d"x = (x^2"e"^(-4x))/(-4) + 1/2 [(-x"e"^(-4x))/4 - 1/16 e"^(-4x)]`

E(X) = `16[(x^2"e"^(-4x))/(-4) - (x"e"^(-4x))/8 - "e"^(-4x)/32]_0^oo`

= `16[0 - ((-1)/32)]`

= `16[1/32]`

= `1/2`

E(X2] = `int_-oo^oo x^2 f(x) "d"x`

= `16int_0^oo x^3"e"^(-4x)  "d"x`

Using integration by parts method

Let u = x3

⇒ du = 3x2 du

And `int "dv" = int "e"^(-4x) "d"x`

⇒ v = `"e"^(-4x)/(-4)`

`int "u" "dv" = "uv" - int "v" "du"`

`intx^3"e"^(-4x) "d"x = - (x^3"e"^(-4x))/4 + 3/4 int"e"^(-4x) x^2 "d"x`

= `- (x^3"e"^(-4x))/4 + 3/4[(x^2"e"^(-4x))/(-4) - (x"e"^(-4x))/8 - "e"^(-4x)/32]`

∵ Using E(X) integration]

= `- (x^3"e"^(-4x))/4 - 3/16 x^2"e"^(-4x) - 3/32 x"e"^(-4x) - 3/128 "e"^(-4x)`

E(X2) = `16[- (x^3""^(-4x))/4 - 3/16 x^2"e"^(-4x) - 3/32 x"e"^(-4x) - 3/128 "e"^(-4x)]_0^oo`

= `16[0 - ((-3)/128)]`

= `16[3/128]`

= `3/8`

Variance Var(X) = E(X2) – [E(X)]2 

= `3/8 - 1/4`

= `(3 - 2)/8`

= `1/8`

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Mathematical Expectation
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 11: Probability Distributions - Exercise 11.4 [पृष्ठ २१०]

APPEARS IN

सामाचीर कलवी Mathematics - Volume 1 and 2 [English] Class 12 TN Board
अध्याय 11 Probability Distributions
Exercise 11.4 | Q 7 | पृष्ठ २१०

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