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- 1.STATISTICAL INFERENCEA continuous random variable X has the distribution function if X 3 Find K and f (x), P.d.f.tion list estion 11 estion 12 estion 13 estion 14 estion 15 estion 16 estion 17 crunch A simple random sample of size n = 53 is obtained from a population that is skewed left with μ=77 and o=4. Does the population need to be normally distributed for the sampling distribution of x to be approximately normally distributed? Why? What is the sampling distribution of x? the sample size increases. OB. Yes. The central limit theorem states that only for underlying populations that are normal is the shape of the sampling distribution of x normal, regardless of the sample size, n. OC. No. The central limit theorem states that only if the shape of the underlying population is normal or uniform does the sampling distribution of x become approximately normal as the sample size, n, increases. OD. No. The central limit theorem states that regardless of the shape of the underlying population, the sampling distribution of x becomes approximately normal as the sample size, n, increases. What is the…
- Example 7-2: Central Limit Theorem Suppose that a random variable X has a continuous f(x)=/2, 4Let {Xn}-1 be a sequence of random variables that converges in probability n=1 Prove that the limiting random variable is unique P-a.s., i.e., if Xn 4 X and Xn 4 Y, then P(X = Y) = 1.Theorem 27 (CENTRAL LIMIT THEOREM (CLT)) Let X1, X2,. .. be independent identically distributed random vari- ables with finite expectation and finite variance. Let S, => X4. 2=1 Then Sn – ESn 30), then the cdf of the rescaled random variable S, is approximately equal to the cdf of a standard-normal random variable Z ~ N(0,1), i.e. Sn – ES, P 3000) (i) with the Markov inequality, (ii) with the normal distribution.poisson distribution3. Let X1, X2,..., be a sequence of independent and identically distributed random variables. Let N be Poisson distributed with mean u and is independent of the X,'s. Define N W =EX;. i=1 We define W = 0 if N = 0. (a) Suppose each X; is normally distributed with mean 0 and variance 1. Work out the moment generating function for W given N. [4 marks] (b) Show that the moment generating function of W is given by Mw (t) = exp(ue"2 - ), t eR. [5 marks] (c) Calculate the mean and variance of W. [5 marks] (d) Now consider Z = NX1. Find the mean and variance of Z. [6 marks]) Let Yhave a binomial distribution with parameters n and p. H,: p= 1 is rejected 1 is accepted if Y2 c. Find n and c to give a power function T(p) 2 and H,:p> so that a = 0.10 and T = 0.95 , approximately.A random sequence X, is defined by X, = A"-K) u[n – K] where A and K are statistically independent random variables, and u[·] is a unit step sequence. Random variable A is uniformly distributed between 0 and 1. Random variable K is a discrete random variable which takes values equal to –1, 0, and 1 with equal probability. (a) Determine the mean sequence µx[n] and plot its values for n= –1,0,1,2,3. (b) Determine the auto-correlation bi-sequence Rx[m, n). (c) Comment on the stationarity of Xn.1.STATISTICAL INFERENCE