6) Consider the random variable Z₁ = cos (2 * π * Yn/n) where Yn is binomial(n,p) a. What is its asymptotic variance for arbitrary p as n increases b. What happens at p=1/2
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![6) Consider the random variable Zn = cos (2 * T
: * Yn/n) where Yn is binomial(n,p)
a. What is its asymptotic variance for arbitrary p as n increases
b. What happens at p=1/2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92acf7d2-b8a7-4489-8f54-64b20b83f80b%2F0c77e7e9-9774-434e-9a83-9b478dcf541f%2Fm7rs42j_processed.png&w=3840&q=75)
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- Identify the distributions of the random variables with the moment- generating functions shown below. For each random variable also indicate what the mean and the variance are. a) m(t) = e^2.2(e^t−1)b) m(t) = 1/(1 − 2t)^2 c) m(t) =( e^5t−e^t )/4tQ Let Z be a standard normal random variable 2. Show that E[Z^*'] = n E [z"-¹]. to compute E [24] Use the resultQ5 Find the variance for the PDF px(x) = e-«/2, x > 0.
- 2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.BapuA time series {yt} follows an MA(2) model: Yt = 2 + Ut +0.54t-1 + 0.4ut-2. Assume that ut is a white noise series with a mean of O and a variance of 2. Please calculate Var(yt) (i.e. the variance of) 2.96 1.41 2.82 O 1.98
- 5. Let X be a positive random variable; i.e., P(X - log(EX) (b) E[log(1/X)] > log[1/EX] (c) E (X³) > (EX)³Suppose W is a random variable with the pdf f(w) = 3u3e-w³/343 for w≥ 0.1 If Y = -12W + 11, what is the pdf of Y? еAssume that X and Y are random variables with the following parameters: 4 аnd o'y 13, 12, о Hy What is the variance of Z if Z = 2X - Y + 1 and ory =0.8
- EL 466 416 13.) The continuous random variable (RV) X is uniform over [0,1). Given Y = -ln X what is P({09.1.1. X, and X2 are iid random variables with variance Var[X]. (a) What is E[X, - X2]? (b) What is Var[X - X2]?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 tSEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. FreemanMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman