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- Let a and b be any real numbers and let x = (x1,Xz -- -- Xp m-, X) be a data set with a sample mean f and a sample variance s. Show that if y, = (ax, + b), then S3 = a*s;6. Let X ~ N(0, 1), the standard Normal distribution. (a) Determine the 3rd moment of X. (b) Determine the 4th moment of X. (c) Let Y = X2. Determine the variance of Y. (d) Find the nth moment of X, where n = 5, 7,9,. .7. Let (r1, 12 ., T„) be a random sample from a normal distribution with mean u and a variance o?. Also let S (x; - )² be the sum of squares of the random sample. %3D State without proof the sampling distribution of the following; (a) t(x) = ao + b;a;, i = 0,1, .., n, a, are positive real constants. i=l n(ī – µ)² (x – 4) (b) t(r) = g2 (f) t(x) = (c) t(x) = (x – 4) (g) t(x) = i=1 (c) t(x)= i=1 i=1 (d) t(r) = E (*; – äm-n)² %3D j=1 i=m+1 (h) t(x) = m - 1 n - m - 1 (x – 4) (e) t(r) =
- ~ N (N, 6) with M. tstiny H. : MS M. Us me a. b. Jf BCM)= 0-2 and Find BUHo+ 6)= 0.95, Find Sample Size, n. %3D P-Value. C.6. Let X ~ N(0, 1), the standard Normal distribution. (c) Let Y = X?. Determine the variance of Y. (d) Find the nth moment of X, where n = 5, 7, 9, . .Let X~ Normal(2, 1) and Y = 2X-3. What is the distribution of Y? a. There is insufficient information. b. Y Normal(1,4) c. Y Normal(1, 1) d. Y Normal(1,-1)
- Q5/ Let X1, ., X,n be a r.s. from population with p. d.f. f(x) = 0, 0> x>0* 0.W. X(1) Let X(1) < .. < X(n) be the order statistics. Find the p. d. f. of U = ) %3D X(n)6. Choose 2 points (X, Y) uniformly at random from the circle. Find the expected area of the rectangle with opposite corners given by X and by Y.X1,., X,Let it be a random sample chosen from a normal distribution with a mean of m and a variance of 10.P (X, – X)² < 44 = 0.025 what is the value of the sample size n? O A. 13 О В. 14 Ос. 11 O D. 12
- X,, X2, .,X, is a random sample selected from the normal distributionx,~N(6,25) and 52 is the sample mean and its variance P(1.860 < 3(X – 6)/S) which of the following is the probability? A. 0.10 B. 0.95 O C. 0.25 D. 0.053. Let X1,.., Xn be a random sample from a uniform distribution on the interval [a, b). In other words, X U(a, b). Obtain method of moments estimators for a and b.X~ N(50, 11). Suppose that you form random samples of 25 from this distribution. Let X be the random variable of averages. Let EX be the random variable of sums. Part (a) Part (b) Part (c) Part (d) Find the 30th percentile. (Round your answer to two decimal places.) Part (e) Sketch the graph, shade the region, label and scale the horizontal axis for X, and find the probability. (Round your answer to four decimal places.) P(46 < X<52) = 0.1059 x Part (f) Sketch the graph, shade the region, label and scale the horizontal axis for X, and find the probability. (Round your answer to four decimal places.) P(12< X<46) = Part (g) Give the distribution of ΣΧ. EX- Part (h) Find the minimum value for the upper quartile for EX. (Round your answer to two decimal places.) Part (1) Sketch the graph, shade the region, label and scale the horizontal axis for EX, and find the probability. (Round your answer to four decimal places.) P(1150SEE MORE QUESTIONS