(b) Consider an urn containing balls labelled 0, 1, 2, ..., n − 1 and the experiment - of drawing n of these balls uniformly and without replacement. Let X, denote the label of the ball drawn in the i-th step, 1 ≤ i ≤n. (i) For any 1 ≤ i ≤ n, what is E[X;] and V[X;]? Justify your answer. (ii) Compute Cov[X₁, X₂]. (iii) Suppose now that n is an unknown parameter and you observe the absolute difference between the labels of the first two balls, that is, Z := |X₁ – X₂|. Can you find an unbiased estimator of n based on Z? Justify your answer.

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A: By the definition of covariance we know,  Cov(X,Y) = E[(X-E(X)) (Y-E(Y)] = E[XY - XE(Y) - YE(X) +…

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A: Xn:n≥1 be independant N0,1 random variables.

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A: Haiyy

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A: denotes a random variable. denote fixed constants. are iid .

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A:

Q: Let Y;, i = 1,...,n be independent N(Bo + YiB1, o²) and let W1,..., Wm be independent N(Y + w;Y1,…

A: @solution:::   X~N(29300,2000) X bar follows typical appropriation with xbar=mu=29300 test…

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A: Given Yt follows the moving average process of order 1 . Yt=εt-θεt-1 where εtis independent and…

Q: 3. Random variables X and Y have joint PDF S Ary 0<I<1,0 < y <1 fxx(1, y) = { 0 otherwise a) What…

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer three sub-parts at a…

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A: Let for  Let ,  Now, Now, Then, Now,

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A: 15(balls)=6(white)+9(red)

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A:

Q: iii) What is the value of the sample size m, if P(|Ỹ — µỹ| ≥10) < 0.04?

A: First we have to determine the mean and variance of Y and then use Chebychev's inequality to get the…

Q: 13. Let {Xn, n ≥ 1} be iid with P[X₁ = 1] = p = 1 - P[X₁ = 0]. What is the probability that the…

A:

Q: Let x1, X2 ... Xn d n(0, 0). That is, o = q² in the usual notation and the likelihood function…

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Q: 7. Let X₁, X2,..., Xn be independent variables, X; being N(µi, 1), and let y = = x² + x² +...+X²/2.…

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Q: Question 4. a) Let X be a random variable and let EX] = µ < o, V[X] = o². Find a random variable Y…

A: Hello thank you for the question, As you have mentioned question number 'a', answering only the…

Q: Let the random variables (discrete) X1, X2 have the joint pmf p(x1, x2) = 1/3, for (x1, 12) =…

A: Covariance is given by, Cov(X1,X2)= E(X1)*E(X2)-E(X1*X2) E(X1)= (summation X1*p(X1,X2)) =…

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Q: Exercise I. Let X,Y~ N(0, 1) be jointly Gaussian random variables and let t€ R. The goal of this…

A: Given::-Let X, Y~ N(0, 1) be jointly Gaussian random variables and let t€ R.

Q: 15. (a) A finite family B₁, ie I of o-algebras is independent iff for every choice of non-negative…

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Q: . Consider a birth-death process with parameters λ = 2.5 · (5 — n) and μn Find the stationary…

A: exact explanation is given below.Explanation:

Q: *In the 1800s, a man with dwarfism who lived in Utah produced a large number of descendants: 22…

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Q: 1. Let X and Y be continuous random variables. The joint PDF of X and Y is | 12.r³y 0<y<I<1, f(r, y)…

A:

Q: Suppose X - N(u, o?) where o is known. Let X1, X2,.... X, be a random sample of size n. Prove that u…

A:

Q: a) Show that for two independent random variable X, Y MX+Y (t) = MX(t)MY (t) b) Suppose X is…

A: Suppose X and Y are Binomially distributed with parameters (n,p) and (m,p) respectively. Let Z = X+Y

Q: 18. Coupling. If X and Y are random variables on (S, B), show sup |P[X EA] - P[YE A] ≤ P[X #Y]. AEB

A: Given::- Coupling. If X and Y are random variables on ( Omega , mathcal B ), show su p A in mathcal…

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A:

Q: 1. Suppose that Y, follows the Moving Average process of order 1 (MA(1)) model Y₁ = &-0₁-1, where is…

A: Suppose Yt follows the moving  average process of order 1 MA(1) model. Yt=εt-θεt-1 where εt is…

Q: -3 Let A = -4 -3 -1 9 Define the Lnead teanofamatia T:IR²> IR° by T2)= AR. Find the inages of ů= [oI…

A: Put the value of u and v in the definition of T

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A: a) To calculate the Cramer-Rao Lower Bound (CRLB) for the parameter A in the given model, we need to…

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Q: Let X₁, X₂, X3,..., Xn be a random sample of size n from population X. Suppose that X~N(0, 1) Σ1Xi…

A: From the given information, Moment of generating function of normal distribution is eμt+σ22t2. For…

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Q: b) Let Zi = X₁−μx ~N(0,1), and W₁ = Yi-Y~N(0,1), for i = 1,2,3,...,10, then: σχ σy i) State, with…

A: Given Zi~N(0,1) and Wi~N(0,1)  Result: t~F(a,b) Then  Mean=E(t)=b/(b-2)…

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Q: 4.9) Let p(k) = P(X = k) = c2 k for k = 1, 2, 3, and p(k) = 0 otherwise. Find the value of c that…

A: We have given that pmf p(k) = P(X = k) = c2 k for k = 1, 2, 3, and p(k) = 0 otherwise

Q: X = r cos(27/8) and Y=rsin(27/8), a discrete uniform random variable on the set {0, 1, 2, ..., 7}.…

A:

Q: 2. Suppose that Yt follows the stationary AR (1) model Yt=2.5+0.7Yt−1+&varepsilon;t, where…

A: The Given stationary AR(1) model yt=2.5+0.7yt-1.