The life expectancy (in years) of a certain brand of clock radio is a continuous random variable with the probability density function below.
[tex]f(x) = \{_{0, otherwise}^{2/(x+2)^2, if x \geq 0}[/tex]
(A) Find the probability that a randomly selected clock lasts at most 6 years.
(B) Find the probability that a randomly selected clock radio lasts from 6 to 9 years.
(C) Graph y = f(x) for [0,9] and show the shaded region for part (A).

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A1peakenbe A1peakenbe · Answer 1 · 2019-08-23T09:08:15+02:00

Answer:

Part a)

Given that

[tex]f(X)=\int_{0}^{X}\frac{2}{(x+2)^{2}}dx[/tex]

Thus the probability that a randomly selected clock lasts for at most 6 years is

[tex]f(6)=\int_{0}^{6}\frac{2}{(x+2)^{2}}dx[/tex]

[tex]f(6)=[\frac{2}{(x+2)}]_{0}^{6}\\\\f(6)=0.75[/tex]

Part b)

The probability that a clock lasts between 6 to 9 years equals

f(9)-f(6)

[tex]f(9)=\int_{0}^{9}\frac{2}{(x+2)^{2}}dx\\\\f(9)=0.818[/tex]

Thus probability becomes 0.818-0.75 = 0.0681

part c )

See attached figure

For shaded region of part a see attached figure