I've tried to solve this series of questions mulitple times, but they end up incorrect.
A) Let T be the amount of years a machine part works. Assume the probability distribution of T is given by:
$$f(t)=\dfrac{t}4 \, e^{-t^2/8}$$
Find the probability that the machine part works beyond 3 years.
B) A similar machine part has already worked for 1/2 year, what is the probability that it will work for at least another 3 years?
C) Use the central limit theorem to find an approximate probability that the average life expectancy of 20 independent components will be at least 3 years.
Hints:
If the density is $f(t)=\dfrac{t}4 \, e^{-t^2/8}$ then the cumulative distribution function is $F(t)=\Pr(T \le t)=\displaystyle \int_{\tau=0}^t f(\tau)\, d\tau=1-e^{-t^2/8}$ for $t\ge 0$
But you are interested in $\Pr(T \gt t)=1-F(t)$. So the answer for (1) is $1-F(3)$ while the answer for (2) is $\dfrac{1-F(3.5)}{1-F(0.5)}$
Question (3) probably needs you to calculate the mean and variance, which may involve more integration