7. Assume that X₁,..., Xn is a random sample from a Bernoulli (p) and let Yn n 1/1 X₁. For p # 1/2, Determine the asymptotic distribution of √n[Yn(1 – Yn) - p(1 - p)].
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7. Assume that X₁,..., Xn is a random sample from a Bernoulli (p) and let Yn n 1/1 X₁. For p # 1/2, Determine the asymptotic distribution of √n[Yn(1 – Yn) - p(1 - p)].
7. Assume that \( X_1, ..., X_n \) is a random sample from a Bernoulli(p) and let \( {Y}_n = \1/n\sum_{i=1}^{n} X_i \). For \( p =/ 1/2), Determine the asymptotic distribution of \( \sqrt{n}({Y}_n(1 - {Y}_n) - p(1 - p)) \).
![7. Assume that X₁, Xn is a random sample from a Bernoulli(p) and let Y₁ =
EX. For p # 1/2, Determine the asymptotic distribution of √Y (1-
Yn)-p(1-p)].
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- Let X₁, X2, ..., Xn be a random sample from a normal population with mean μ and variance o² taken independently from another random sample Y₁, Y2, ..., Ym from a normal population with mean Ⓒ and variance 0². Let Z₁ = Xi- Derive the distribution of T = z² o (Y₁-P) 772-1Let X1, . . . , Xn be iid from Exp(β). Find the distribution of the sample maximum Y = X(n). Is this an exponential distribution? (This question was rejected this morning. I did double check and there is no missing info with this question.)Let X1, . . . , Xn a random sample from a Uniform distribution on the interval [0, 2θ + 1].(a) Find the MLE of θ.(b) Find the MLE for the variance of the distribution of Xi.
- Let Y₁, Y2, ..., Yn denote a random sample of size n from a population with a uniform distribution on the interval (0,0). Consider = Y(1) = min(Y₁, Y₂, ..., Yn) as an estimator for 0. Show that is a biased estimator for 0.6. Let (y1, Y2, ., Yn) be a random sample from the uniform distribution on the interval [A – av5, 1 + av/5], where A is the unknown parameter and a is a known constant. Determine the moment substitution estimator of d.5. let X1 and X2 have the joint pdf f(x1,x2) = x1+ x2, 0 < x1 < 1, 0 < x2 < 1. Find the conditional mean and variance of X2 given X1 = x1, 0 < x1 < 1.
- Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).Consider a random sample from NB(r, p) where the parameter r is known to be 3. An experiment is run with n = 10 trials and the sample mean is observed to be x̄ = 0.6. (a) Derive a formula for the MLE p̂ as a function of n, r and X̄ . (b) Find the estimate of p. (c) Find the MLE for the population mean.The Volatility X for the S&P stock index on a given day is a normal random variable with mean = 10 and standard deviation = 2 The volatilities recorded over a 100-day period on the S&P500 are Y1, Y2, ... Y100. Assume that these Yi's are independent and identically distributed, uniform on the interval [5,15]. Let V = (Y1 + Y2 + ... + Y100)/100. What approximately is P[9.5 < V < 10.5]?