. I have found the first part to the first question which is .22 with the formula p1+p1*p2 and it is equal to the second part. However I'm not sure how to find the third part.
. For the second question I was told to use gamma distribution but I have the integral from 0 to infinity e^((-10+t) µ) µ^2 500 du
.The third question I am unable to solve so any help would be greatly appreciated.
For the first question, use the hint. $K_{90} = \lceil T(90) \rceil$, thus $$\Pr[K_{90} \wedge 2 = k] = \Pr[\min(\lceil T(90) \rceil, 2) = k]$$ where $T(90)$ is the future lifetime random variable for $(90)$. If $(90)$ survives even an infinitesimal amount of time, then $T(90) > 0$, hence $K_{90} \ge 1$. But with probability $q_{90} = 1 - 0.2 = 0.8$, $(90)$ survives less than $1$ year, so $\Pr[(K_{90} \wedge 2) = 1] = 0.8$. And since $$p_{90} = 0.2, \quad p_{91} = 0.1$$ implies $$p_{90} q_{91} = 0.2 - 0.1 = 0.1.$$ This means the probability $(90)$ survives one year but not two is $0.1$, which corresponds to $1 < T(90) \le 2$. Therefore $$\Pr[K_{90} = 2] = 0.1.$$ For subsequent values of $k$, the condition is never satisfied because $\min(K_{90}, 2) = 2$ for $K_{90} > 2$. So we have $$\operatorname{E}[K_{90} \wedge 2] = 1 \cdot \Pr[K_{90} = 1] + 2 \cdot \Pr[K_{90} = 2] = 1(0.8) + 2(0.1) = 1.$$
For the second question, the force of mortality has density function $$f_\mu(u) = 500 u^2 e^{-10u}.$$ Being a hazard rate, this is the density $f_{T(30)}(t)$ of the future lifetime random variable $T(30)$ divided by the survival function $S_{T(30)}(t) = {}_t p_{30}$. That is to say, $$f_\mu(t) = \frac{d}{dt} \left[ - \log {}_t p_{30} \right],$$ and $${}_t p_{30} = \exp\left(- \int_{u=0}^t f_\mu(u) \, du\right).$$ We can perform the integration by parts; the corresponding indefinite integral is $$-\int y^2 e^{-y} \, dy = (y^2 + 2y + 2) e^{-y} + C.$$ The result is rather strange but it is consistent with the provided force of mortality function.
The third question makes no sense to me. What is $f_X(t)$ supposed to be? Is it a probability mass function? Is it a density? If so, then for what random variable?