Question

Losses covered by a flood insurance policy are uniformly distributed on the interval (0,2). The insurer pays the amount of the loss in excess of a deductible d. The probability that the insurer pays at least 1.20 on a random loss is 0.30. Calculate the probability that the insurer pays at least 1.44 on a random loss.

Answer 1

Answer:

The probability that the insurer pays at least 1.44 on a random loss is 0.18.

Step-by-step explanation:

Let the random variable X represent the losses covered by a flood insurance policy.

The random variable X follows a Uniform distribution with parameters a = 0 and b = 2.

The probability density function of X is:

$f_{X}(x)=\frac{1}{b-a};\ a<X<b\\\\\Rightarrow f_{X}(x)=\frac{1}{2}$

It is provided, the probability that the insurer pays at least 1.20 on a random loss is 0.30.

That is:

$P(X\geq 1.2+d)=0.30$

⇒

$P(X\geq 1.2+d)=\int\limits^{2}_{1.2+d}{\frac{1}{2}}\, dx$

                $0.30=\frac{2-1.2-d}{2}\\\\0.60=0.80-d\\\\d=0.80-0.60\\\\d=0.20$

The deductible d is 0.20.

Compute the probability that the insurer pays at least 1.44 on a random loss as follows:

$P(X\geq 1.44+d)=P(X\geq 1.64)$

                        $=\int\limits^{2}_{1.64}{\frac{1}{2}}\, dx\\\\=|\frac{x}{2}|\limits^{2}_{1.64}\\\\=\frac{2-1.64}{2}\\\\=0.18$

Thus, the probability that the insurer pays at least 1.44 on a random loss is 0.18.