Consider a random variable X that follows a Beta distribution with E[X] : Find the method of a+1"

f (x) = 4xc for 0 ≤ x ≤ 1 ; Find the constant c so that f (x) is apdf of some random variable X, and then find the cdf, F (x) = P(X ≤ x). Sketch graphsof the pdf f (x) and the cdf F (x), and find the mean μ and variance σ2.

With an American penny, the likelihood of getting H when it is spun on edge is 0.3. If X is the random variable where X(H ) = 1, X(T ) = −1, find the expected value E(X), the variance, Var(X), and express X in its standard form.

Assume the lifetime of electronic systems T (based on months) has Exponential distribution with λ=0.01. We select a random sample of 5 systems (random lifetime of these systems are T1, T2, T3, T4, and T5). Assume, S= T1+T2+ T3+ T4+ T5. a. Find E[S] . b. Find Standard Deviation of S (σSσS). c. Find median of T1.