Exponential random variables

Since the mean duration of a solar panel is 4 months and the duration is an exponential random variable we know that the density function for failure time (with time given in months) is given by:

$f(x)=\frac{1}{4}e^{-\frac{1}{4}x}$. We can be saved some extra work by knowing the cumulative distribution function as well $F(x)=1-e^{-\frac{1}{4}x}$.

The odds of a given panel failing in 14 months are therefore $F(14)-F(0)$, since $F(0)=0$ the odds of a panel failing are $F(14)$.

The sattellite only fails if the three independent panels fail, so the odds of the sattelite failing in 14 months are $F(14)^{3}$, i.e. each panel fails within 14 months. The probability of the sattelite remaining operational after 14 months is therefore the complement of the probability of the sattelite failing, $1-F(14)^3$ with F defined as earlier.