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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
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Solution
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
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