Let X1, X2, . .., Xn be a random sample from a population with a pdf a+2022 (a+ 2021) () ; х > 0, а > 0 x+1 f(x) = 0, elsewhere. n Show that ||| is sufficient for a. Xi +1

Show that for any random variable E[X2]≥ (E[X])² and When does one have equality?

Suppose W is a random variable with the pdf f(w) = 3u3e-w³/343 for w≥ 0.1 If Y = -12W + 11, what is the pdf of Y? е

Suppose that the waiting time X (in seconds) for the pedestrian signal at a particular street crossing is a random variable with the following pdf. · (1 − x/71)² 0 ≤ x < 71 {0 f(x) = {√ √ otherwise If you use this crossing every day for the next 6 days, what is the probability that you will wait for at least 10 seconds on exactly 2 of those days?