7. Radar returns (power) are modeled in terms of the following (gamma) density, x f(x) == e ¹U (x) (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.

Example: The Gamma Distribution Suppose X ~ T(a, B), then M(t) = dx BaT(a) 22 Now we do a little high school algebra. Define & = Then one may write 1-Bt" Equation (22) as: M(t) = ()ª / (23) dx One recognizes the integrand of Equation (23) as a r(a, 6) density, hence 8. M(t) = (Gr"(1) = (1 - AY• (24) (1 – Bt)a Exercise 5. Verify the algebra leading from Equation (22) to Equation (23).

сх 4. Elongation (in percent) of treated metal plates are random with probability density function: f (x) = ,5 < x < 25 600 Find the value of c which makes f(x) a properly defined pdf. Find the variance of elongation. а. b. Find the probability that the elongation is between 10% and 15% or between 12% and 20%. That is if E is the random variable representing elongation, find P(10 < E < 15 or 12 < E < 20). с.

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.