The lifetime, in years, of a certain type of fuel cell is
a random variable with probability density function
f (x) = { 81 / ( x + 3 ) ^ 4 , x > 0
0 , x ≤ 0
a. What is the probability that a fuel cell lasts more
than 3 years?
b. What is the probability that a fuel cell lasts between
1 and 3 years?
c. Find the mean lifetime.
d. Find the variance of the lifetimes.
e. Find the cumulative distribution function of the
lifetime.
f. Find the median lifetime.
g. Find the 30th percentile of the lifetimes.
1 Answer
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a. P(X > 3) = integral [81 / ( x + 3 ) ^ 4] evaluating x from 0 to 3
b. P(1 < X < 3) = integral [81 / ( x + 3 ) ^ 4] evaluating x from 1 to 3
c. E(X) = integral {x * [81 / ( x + 3 ) ^ 4]} evaluating x from 0 to positive infinity
d. V(x) = E(X^2) - [E(X)]^2. We already have E(X) in #3, so you just have to square it. To get E(X^2), integral {x^2 * [81 / ( x + 3 ) ^ 4]} evaluating x from 0 to positive infinity
e. P(X < x) = integral [81 / ( x + 3 ) ^ 4] evaluating x from 0 to x