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Question
The probability density function of X is given by
`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`
Find the distribution function
Solution
The distribution function F(x) = `int_-oo^x f(u) "d"u`
Case 1:
x < 0, F(x) = `int_-oo^x f(u) "d"u` = 0
Case 2:
x ≥ 0, F(x) = `int_-oo^x f(u) "d"u`
= `int_-oo^0 f(u) "d"u + int_0^x f(u) "d"u`
= `0 + k int_0^x "e"^((-u)/3) "d"u`
= `1/3 [("e"^((-u)/3))/((-1)/3)]_0^x`
= `- ("e"^((-x)/3) - 1)`
= `1 - "e"^((- x)/3)`
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