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

Answer 1

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