Reality Check: Benefit paid by any particular insurer cannot be negative, irrespective of the deductable.
You seek the probability for: $$\{\max(X-1,0)+\max(Y-2,0)\leq 5\}$$
Partitioning on indication that $Y\leq 2$ or not:$$\{Y\leq 2\,,\max(X-1,0)\leq 5\}\cup\{Y>2\,, \max(X-1,0)+Y\leq 7\}$$
Partitioning again on indication that $X\leq 1$ or not:$${\{X\leq 1\,, Y\leq 2\}}\cup{\{1<X\leq 6\,, Y\leq 2\}}\\\cup{\{X\leq 1\,, 2<Y\leq 7\}}\cup{\{X>1\,, Y>2\,, X+Y\leq 8\}}$$
How can X be any number greater than 1 if Y is less than or equal to 2 and $X+Y$ be $<8$ ?
In the last part, we require $1<X$ and $2<Y$ and $X+Y\leq 8$, which is to say, the probability for $1<X\leq 6$ and $2<Y\leq 8-X$:
$$\mathsf P(1<X\leq 6\,, 2<Y\leq 8-X) = \int_1^6\int_2^{8-x}\tfrac 1{100}\mathop{\rm d}y\mathop{\rm d}x = \tfrac{1}{8}$$
Do similar for the other parts.
Given the memoryless property of the exponential distribution, the distribution of $Y=\max(X-d,0)$ will have the same $\lambda=\frac 1 {10}.$
The expected probability that the insurance company will have to pay at all (claim larger than deductible) is given by the distribution of the exponential evaluated at $x>d$:
$$\Pr(X>d)= e^{-0.1 d}$$
Therefore $Y=0$ with probability $1-e^{-0.1d},$ and follow an exponential distribution with mean $10$ with probability $e^{-0.1 d}.$
The $E(Y)=0\cdot (1-e^{-0.1d}) + 10 \cdot e^{-0.1d}$
We are going to need the formula $\text{Var}=E(Y^2)-E(Y)^2,$ and the latter part after the minus is $100\cdot e^{-0.2d}.$
The $E(Y^2)$ requires some leg work. By LOTUS,
$$\begin{align}
E(Y^2) &= \int_0^\infty y^2 \cdot 0.1 e^{-0.1y}\;dy\\[2ex]
&=\left.y^2 \cdot (-e^{-0.1 y})\right|_{0}^{\infty} - \int_0^{\infty} (-e^{0.1 y}) \cdot 2y \; dy\\[2ex]
&=\frac{2}{0.1}\int_0^{\infty}y\; 0.1e^{0.1 y} \; dy\\[2ex]
&=\frac{2}{0.1}E(Y) = \frac{2}{0.1}\frac{1}{0.1}=2\cdot 100
\end{align}$$
but before we apply the formula of the variance, we need to go back to the deductible, and the probability that the company actually has to make payments:
$$E(Y^2)=0\cdot (1-e^{-0.1d})+200\cdot e^{-0.1d}.$$
Now we obtained the desired result $100 \left(2 e^{-0.1d}- e^{-0.2d}\right).$
Best Answer
Let Y'=Unreimbursed portion and X=the total loss incurred. Y' =144 =
X, for X<180 or 180,for X>180.144 = integral of (X/b)dx+180*prb(X>180)dx with limit 0 to 180. prb(X>180) = (b-180)/b, this is done using CDF of the uniform distribution.Doing the integration and evaluation gives b=450