Let {x .,x } is a random sample coming from N(μ, σ^), and suppose σ is known. (a) Show that hat(e) = x n 2 is an unbiased estimator for u. (b) Find the efficiency of hat(e). ง C 5. Let {X1, X2,. ,Xn} is a random sample coming from N(μ, o²), and suppose σ² is known. .... (a) Show that = x² - 2 is an unbiased estimator for μ². (b) Find the efficiency of Ô.

Suppose X~ N(2,6). What value of a has a z-score of three? x =

Let X be a random variable with the function, -(x -A) "for x > 2, Let f (x) = { elsewhere. Derive u and show that T = X is biased estimator of 2 . Can you modify T =X in order to get an unbiased estimator of 2.

A random variable Y has a uniform distribution over the interval (8,, e,). Derive the variance of Y. Find E(Y)? in terms of (e,, e,). E(Y)? = Find E(Y) in terms of (e,, e,). E(Y) = Find V(Y) in terms of (8,, e,). V(Y) =

let x be a random variable with moment generating function Mx(t)=(0.6 + 0.4e^t)^20 then the variance of x is

Suppose we have random samples (X₁, X2, X3, X₁, X5) and it is known that X; Exponential (0). Th observations are (x1, X2, X3, X4, X5)=(0.32, 2.37, 1.26, 2.18, 1.8) When typing in formulae, use the notation below xi theta x_i When typing in formulae, don't forget to use "*" for multiplication Likelihood Find the likelihood function by filling in the gaps below. Express the Likelihood functions through and X1, X2, X3, X4, X5. 5 = The formula for calculating the likelihood is L(0; x) = [] ƒ(X; ; 0); i=1

3. The length of time required by students to complete a 1 hour exam is a random variable with a pdf given by: f (x) = = ca + for 0 sæ<1 a. Find c. Enter c as a reduced fraction. C = b. Find F(x). Enter coefficients as reduced fractions and use ^ to denote powers. F(x) = c. Find the probability that a student takes less than 30 minutes to complete the exam. Enter the probability as a reduced fraction. prob = d. Find the median length of time to take the exam. Enter your answer in hours with 4 decimal places. median = hours e. Find the length of time for the first 10% of the students to complete the exam. Enter your answer in hours with 4 decimal places. hours answer = f. Find the expected value, variance, and standard deviation of X. Enter your answer in hours with 4 decimal places (or hours squared in the case of variance), hours expected value =

We know that the point estimator o` is an unbiased estimator for the parameter 0 if E(e') = 0.If the estimator is biased (not unbiased), then the difference E(O) – 0 is called the bias of the estimator e and is denoted by Bias(e'). That is,

2.2 Explain how to generate from the Beta (1, 3) distribution using the inverse- transform method.

show that 1-2% show that B(wn) p(ntt, n+t) =표산에.