Suppose that Y ~ Normal(0, 1) and define new random variable X by X=Y². Compute ELYI
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![Suppose that
Y ~ Normal(0, 1)
and define new random variable X by X=Y2. Compute
E[X].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F457222b6-3da3-440b-a3f4-146e4d50c2f1%2F224be14a-6601-424f-836d-ff7b389b7e70%2F1194ch_processed.png&w=3840&q=75)
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- a) Let X₁, X₂, X3,..., X, be a random sample of size n from population X. Suppose that X~N(0, 1) you and Y = -√n. VR i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal rando... variable. iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X₁, X₂, X, and Y₁, Y₂.... Y be random samples from populations with moment generating functions Mx, (t) = eat+t² and My (t) = (-), respectively. 25 i) Find the sampling distribution of the statistic W = X₁ + 2X₂ X3 + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)2> 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-Hy ≥10) <0.04? QUESTION ASuppose that discrete random variable X~uni[1,2,3] A random sample of n=36 is selected from this population. Find the probability that the sample mean is greater than 2.1 but less than 2.5, assuming that the sample mean would be measured to the nearest tenth.Show that two random variables X, and X, with joint pdf. Sx, x, (*1 »*2) = 16 Xl< 4, 2< X2<4 are independent and orthogonal.
- 3. If X;~N(0,1), i = 1,2,3,4 are independent Normal random variables and Y is given as Y =(X,+X,)' +(X, + X,)' . If aY~x², . find the constant a. 2Two random variables X and Y have joint char- acteristic function l@,,m)=exp(-20,?-80;) (i) Show that X and Y are zero mean ran- dom variables.Help please and how to solve
- 3 Find the moment of the random variable X in terms of its characteristic function Фx (@)24. Gaussian random variables X, and X,, for which, X, = 2, o = 9, X, =-1, ox, = 4 and Cxx, =-3, are transformed to new random variables Y, and Y, according to जापर डेरें| Y, =-X, + X, Y, =-2X,- 3X, (iii) Cy,Cy, .. FindGiven the Rayleigh random variable from the density function -(x-a)² Show that the mean E[X] =a +₁₁ 4 ƒ(x) = ²/√(x-a)e 6 b u(x-a) LOS
- Let X, X2, ... , X, be i.i.d. standard normal random variables, i.e., with mean 0 and variance 1, and let Y, = X? for all i = 1, ... , n. Find the moment generating function of Y, to show that Y; ~ x²(1)3. An urn contains exactly 10 red chips and 6 blue chips. Five chips are randomly selected without replacement. Let X denote the number of blue chips in the sample of five chips. (a) What type of random variable is X? Indicate both the type and the appropriate parameters using the "~" notation. Write down the pmf of X, and do not forget to indicate the range of values that x can take on. X ~ px (x) (b) What is the probability that there are between two and four blue chips among the five chosen chips? Express your answer as a decimal. = (*)*Let x1, x2, .. xn be independent standard normal random variables, where n = 9. What is the probability that 3 of them have values in the range [1, 2]? This is how alpha is computed for some sentizing rules. Use 7 decimal places.
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