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Question
The number of miles an automobile tire lasts before it reaches a critical point in tread wear can be represented by a p.d.f.
f(x) = `{{:(1/30 "e"^(- x/30)",", "for" x > 0),(0",", "for" x ≤ 0):}`
Find the expected number of miles (in thousands) a tire would last until it reaches the critical tread wear point
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Solution
We know that,
E(x) = `int_(-oo)^oo x "f"(x) "d"x`
= `int_0^oo x(1/30 "e"^((-x)/30)) "d"x`
= `1/30 int_0^oo x"e"^((-x)/30) "d"x`
= `1/30 [(1!)/(1/30)^(1 + 1)]` ......`[("Using Gramma Integral"),(int_0^oo x^"n""e"^(-"a"x) "d"x = ("n"!)/("a"^("n" + 1)))]`
= `1/30 [1/(1/30)^2]`
= `1/30[1/((1/900))]`
= `1/30 [900]`
= 30
= E(x)
= 30 thousands miles
E(X) = 30,000 miles
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