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- If the PDF of X is f(x)=2x/k2 for 0<x<k, for what value of k is the variance of X equal to 2?1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.Suppose X1, . . . , Xn are i.i.d. from a continuous distribution with p.d.f. fθ(x) = 1/θ if 0 ≤ x ≤ θ, where θ > 0 is an unknown parameter. (a) Find E(X1) (b) Find the MME for θ. (c) Compute the variance of your MME from part (a).
- 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 ??(?) = ??−???(?).Let X be a random variable with mean μ and variance _2. Show that E[(X − b)2], as a function of b, is minimized when b = μ.A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.33. A dietician is asked to test this claim and finds that a random sample of 24 servings has a variance of 1.37. At α=0.01, is there enough evidence to reject the manufacturer's claim? Assume the population is normally distributed. Complete parts (a) through (e) below. (a) Write the claim mathematically and identify H0 and Ha. A. H0: σ2≤1.33 (Claim) Ha: σ2>1.33 B. H0: σ2≠1.33 Ha: σ2=1.33 (Claim) C. H0: σ2≥1.33 Ha: σ2<1.33 (Claim) D. H0: σ2=1.33 (Claim) Ha: σ2≠1.33 (b) Find the critical value(s) and identify the rejection region(s). The critical value(s) is(are) enter your response here. (Round to two decimal places as needed. Use a comma to separate answers as needed.) Choose the correct statement below and fill in the corresponding answer boxes. A. The…