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- Suppose that X1, . . . , Xn is a random sample from the Normal distribution N (0, σ2 ) with parameter σ > 0, Find the Maximum Likelihood Estimation of σ.Suppose Y is a continuous random variable drawn from the uniform distributionon the interval [3, 4], that is, Y ∼ Uniform([3, 4]). Conditioned on Y = y, a second randomvariable X is drawn from the uniform distribution on the interval [0, y]. What is fX(x), thepdf of X?Suppose a random variable Y uniformly distributed over the interval [2, 7]. 1. What is f(y)?
- Show that there are no values of c such that f(x) = cx can serve as the values of the probability distributionof a random variable with the countably infinite rangex = 1, 2, 3, ...If X1 and X2 are independent random variables hav-ing binomial distributions with the respective parameters n1 and θ and n2 and θ, show that Y = X1 + X2 has thebinomial distribution with the parameters n1 + n2 and θ.Suppose that X is a random variable with the density:f(x) = ( θxθ−1 if x ∈ (0, 1) 0 if otherwise The parameter θ > 0 is to be estimated. Suppose that x1 = 0.3, x2 = 0.6 are the sample data. Get the estimate of θ(a) using the Moment Method;(b) using the Maximum Likelihood Method.
- Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]If X1, X2, ... , Xn are independent random variables having identical Bernoulli distributions with the param-eter θ, then X is the proportion of successes in n trials, which we denote by ˆ . Verify that(a) E()ˆ = θ;(b) var()ˆ = θ (1 − θ )n .If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?