In each of tthe following cases, Y is a Gaussian random variable. Find the expected value μ = E[Y]. a) Y has standard deviation σ = 10 and P[Y ≤ 10] = 0.933 b) Y has standard deviation σ = 10 and P[Y ≤ 0] = 0.0.067 c) Y has standard deviation σ and P[Y ≤ 10] = 0.977 (Find μ as a function of σ.) d) P[Y > 5] = 0.5

Consider a random sample X₁, X2,..., Xn from the pdf f(x) = (1 + 0x) −1≤x≤1 where -1 < 0 <1 (this distribution arises in particle physics). Show that = 3X is an unbiased estimator of 0.

Suppose the random variable, X, follows a Bernoulli distribution with parameter 0. The probability function of X is given by p(x; 0) = 0² (1-0)¹-² if x = 0, 1, otherwise. 0 Let X₁, X2,, X₁ be a random sample of size n from the population of X. Assume that the X's are independent and identically distributed with the same parameter. (a) Based on the joint probability function of the sample, state the likelihood function of the parameter. Discuss the difference between the joint probability function and the likelihood function. (b) Show that the sample mean is the maximum likelihood estimator (mle) of 0. (c) Find the method of moments estimator (mme) of 0.

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.