Question

The time to failure in thousands of hours of an important piece of electronic equipment used in a manufactured DVD player has the density function
f(x) = `{{:(2"e"^(-2x)",",  x > 0),(0",",  "otherwise"):}`
Find the expected life of this piece of equipment

Answer 1

Let X be the random variable

`"E"(x) = int_(oo)^oo x"f"(x)  "d"x`

`"E"(x) = int_0^oo x[2"e"^(2x) "d"]x`

= `2int_0^x x"e"^(-2x) "d"x`  ......`{int_0^oo x^"n""e"^(-oox)  "d"x = ("n"!)/(oo^("n" + 1))}`

= `2[(1!)/(2)]`

= `2[1/4]`

= `1/2`

∴ `"E"(x) = 1/2`

Here n = 1, `oo` = 2