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Question
The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function
`f(x) = {{:(3"e"^(-3x), x > 0),(0, "eleswhere"):}`
Find the expected life of this electronic equipment
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Solution
`mu = "E"("X")`
= `int_0^oo x f(x) "d"x`
= `3 int_0^oo x"e"^(-3x) "d"x`
u = x, `int"dv" = int"e"^(-3x)"d"x`
du = dx, v = `"e"^(-3x)/(-3)`
`int"udv" = "uv" - int"vdu"`
= `3[(x"e"^(-3x))/(-3) + int"e"^(-3x)/3 "d"x]_0^oo`
= `3[- (x"e"^(-3x))/3 - "e"^(-3x)/9]_0^oo`
= `3[0 - (0 - 1/9)]`
= `1/3`
Expected life of the electronic equipment = `1/3`
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