1. Suppose that the random variable X has the pdf given by T(a+B) pa-1(1 – x)ª-1 Г() f(x) = for 0 < x < 1 and 0 otherwise. Derive E(X).
Q: 2.5.8) Suppose that Y is a continuous random variable whose pdf is given by f(x) = {K (4y = 2…
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Q: Let X be a continuous random variable with PDF 1 fx(x) for all a E R V27 and let Y = X². Find fy(y).
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Q: 7. Let M(t)=(1/6)et +(2/6)e²t+(3/6)e³t (Moment generating function of X) a. Find E(x) b. Find…
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Q: If E(X) = 6 and E[(X) (X-1) ] = 51, find the second moment= 1 and the variance of X= 2
A: Given Data: E(X)=6 E[(X) (X-1)]=51
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Q: 3) Let X be a continuous random variable with PDF x20 fx(x) ={0 otherwise Find E[X], Var[X] and M…
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Q: ). If X is a continuous random variable whose pdf is f(x): (a) 0 (b) 1 (c) 1/4 (d) 4 (e) None of the…
A: 0 \)" data-mce-style="cursor: default;">f(x)=c(1+x)4;x>0
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Q: Let f(x) = In[x4(x +9)6(x2 + 4)3]. Find f'(x). f'(x) =
A: Given,
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A: Note: Hi there! Thank you for posting the questions. As there are multiple sub parts, according to…
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Q: If the moment-generating function of X is M(t) = {e¹ + find the mean, variance, and pmf of X. 2t
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Q: 1. Let the random variable Y have pdf as f(y\A) = e®=e=t(e-1), y>0,1>0. Show that W = e – 1] ~ x²…
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Q: 13. Find the MGF of a random variable whose moments are u, = (r +1)! 2". Ans Mx(t) = (1-2r) 14. If a…
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Q: - 3 Let f(x) = 8xVæ + x2. f'(x) =
A: Use the derivative for finding the answer.
Q: 9. Show that if t is a consistent estimator of e, then ť is a consistent estimator of 0°.
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Q: Let f(x) = In(x² – 14x + 52) f'(x) =
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Q: Find (f1)'(3) given the following table of values. f(z) f'(z) 3 1
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Q: =e constant K when x > 0.) c so that c. /X₁ X₂)
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- a) You throw two fair dice labeled 1, . . . , 6 independently of each other and let X be the minimum of the two results. Calculate E[X2]. (b) Suppose that X has pdf given by f(x) = 3x^(−4) if x > 1 and by f(x) = 0 if x <= 1. Calculate the median of X.Imagine you have two estimators X and Y, for unknown parameter µ. Assume that they are estimated from independent datasets. Let Z = }(X+Y) be the average of these two estimators. To write the symbol µ, you can write mu. For other math notation, you can E[X], Var(X) and (X+Y)/2 or (1/2)(X+Y). Part a - If X and Y are unbiased estimators, E[X] = E[Y] = µ, then is Z unbiased? Explain why or why not. Your explanation should include a short derivation. Part bL Assume Var(X) = Var(Y). Is Z a lower variance or higher variance estimator than X? Explain your answer. Your explanation should include a short derivation.1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?
- 7. a) Suppose that X is a uniform continuous random variable where 04. Let X be a positive random variable (i.e. P(X 1/E(X) (b) E(-log(X)) > -log(E(X)) (c) E(log(1/X))> log(1/E(X)) (d) E(X³) > (E(X))³Let X ~ F(a, ß). If E(X)= % and E(X²)= a+a² If â = nX? find the method of (n-1)s² | .3 moments estimator ofIf E[X] = 1 and Var(X) = 5, use definition of variance and properties of expectation to find (a) E[(2 + X)^2] (b) V ar(aX) for any constant a. (c) V ar(X + b) for any constant b. (d) V ar(4 + 3X)2.11 Let X have the standard normal pdf, fx(x) = (1/√√2π) e¯2²/² (a) Find EX² directly, and then by using the pdf of Y = X2 from Example 2.1.7 and calculating EY. (b) Find the pdf of Y = |X|, and find its mean and variance.3) Assume that a random variable X has moment generating function as follows: e' + e²¹ M(t)=4-2e¹ ∞ Calculate the mean and variance of X. for tRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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