1. Find the density of Y = 1- e-ax when X ~ Exp(X). Note that the density of X is if r > 0, , otherwise. de-dz f(x) = { What is the name of the distribution of Y?

2. An investor's losses follow a distribution with density function, f(x) =, x > 1 and 0 everywhere else, To recover her losses, the investor purchases an insurance policy with a deductible of $1. Show that the insurance company will go bankrupt. Hint: The expected amount of the losses paid by the company is too large.

The bearing capacity of the soil under a foundation is measured to range from 200 kPA to 450 kPA. The probability density within this range is given by engineering office as f(x) = 3 ,200< x < 450 450 ,otherwise If you know that the column on which stands is designed to carry a load 275 kPA, what is the probability of the failure of the foundation? Do you think redesigning project is necessary?

7. Radar returns (power) are modeled in terms of the following (gamma) density, x X f(x) = ² ² e ²³ U (x) B (a) What is the value of B so that f(x) is a valid pdf? (b) What is the probability that the received power exceeds 2 units. (c) What is the probability that the received power is less tha 6 units. (d) What is the probability that the received power lies between 2 and 6 units (e) What is the probability that the received power is exactly 5 units. (f) If past observations have shown that the received power is always more than 3 units, what is the probability that the the received power will be less than 6 units. (g) What is theprobability that X²>4? (h) What is the probability that sqrt(X)>2?

33.-

Please do only defg