Let X, have a gamma distribution with parameters n/2 and 2 , and let Y, = log X, .Answer the following questions. (X, – n)/ /n converges in distribution to a normal distribution with mean 0 and variance Fill in the blank. v Choose... 1 Vn(Y, – log n) converges in distribution to a normal distribution with mean 0 and variance Fill in the 4 blank. 5 2
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- The moment generating function can be usedto find the mean and variance of the normal distribution.a. Use derivatives of MX(t) to verify that E(X)=meanand V(X) =varianceb. Repeat (a) using RX(t)=ln[MX(t)], and comparewith part (a) in terms of effort.Let X and Y are independent and both have exponential distribution,Find ??(?) , if Z=X+Y. Where PDF of X and Y are given as,??(?) = ??−???(?) and ??(?) = ??−???(?).For a binomial distribution with parameters n=25 and p=0.70, determine the continuity correction for normal approximation for: a) P( X = 20) b) P( X > 20) c) P(X ≤ 20) d) P(X≥ 20)
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- Consider the continuous random variable X = the weight in pounds of a randomly selected newborn baby born in the United States last year. Suppose that X can be modeled with a normal distribution with mean μ = 7.57 and standard deviation σ= 1.26. If the standard deviation were σ = 1.06 instead, how would that change the graph of the pdf of X?For an exponential random variable (X) having θ = 4 and pdf given by: f(x) = (1/θ)e^(−x/θ ) where x ≥ 0, compute the following: a) E(X). b) Var(X). c) P(X > 3).Let X denote 0.025 × the ambient air temperature (˚C) and let Y denote the time (min) that it takes for a diesel engine to warm up. Assume that (X, Y) has joint probability density function f(x,y) = 1.6x (1 − x)(6 + 5x − 4y), for 0 < x < 1, 0 < y < 0.5. While you cannot guess the value of the correlation from the regression curve for X or Y, do they suggest whether it likely is positive or negative?