N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e.,...

Question

Question

N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1 with probability 1/2.

Answer 1

Answer #1

Page No. Date: Solutin Cs) Let ZNNU,L This distibution is called chi-squdrc alth dt PC1X<)=PC<4 - Pr P> Oi30 17 0.2S e2

Answer 2

Similar Homework Help Questions

Let X ~ N(0, 1), and let Z ~ Unif{-1, 1} (i.e. P(Z = -1) =...

Let X ~ N(0, 1), and let Z ~ Unif{-1, 1} (i.e. P(Z = -1) = P(Z = 1) = 1/2) be independent of X. Let Y = ZX. What is the distribution of Y? Show that X and Y are uncorrelated. Are X and Y independent?
2. The chi-square distribution plays a significant role in performing inference on the as- sociation between...

2. The chi-square distribution plays a significant role in performing inference on the as- sociation between categorical random variables (e.g., car injury severity and seat belt usage). If Z ~ N(0,1), then W = Z2 ~ xỉ – that is, W has a chi-square distribution with 1 degree of freedom. Furthermore if Z1, Z2, ..., Zn N(0,1), then W = Z+Z2+...+22 has a chi-square distribution with n degrees of freedom. Here are some helpful facts. Let t > 0 •...
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . ....

4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...

Let X,, X,,... be independent and identically distributed (iid) with E X]< co. Let So 0, S,X, n 2 1 The process (S., n 0 is called a random walk process. ΣΧ be a random walk and let λ, i &gt...

Let X,, X,,... be independent and identically distributed (iid) with E X]< co. Let So 0, S,X, n 2 1 The process (S., n 0 is called a random walk process. ΣΧ be a random walk and let λ, i > 0, denote the probability 7.13. Let S," that a ladder height equals i-that is, λ,-Pfirst positive value of S" equals i]. (a) Show that if q, then λ¡ satisfies (b) If P(X = j)-%, j =-2,-1, 0, 1, 2,...
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with...

4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with probability density function S(*;ð) - ma t?e-vor x>0, zero otherwise. Recall: W=vX has Gamma( a -6, 0-ta) distribution. Y=ZVX; = Z W; has a Gamma ( a =6n, = ta) distribution. i=1 E(Xk) - I( 2k+6) 120 ok k>-3. 42 S. A method of moments estimator of 8 is 42.n 8 = h) Suggest a confidence interval for 8 with (1 - 0) 100%...
Show that if Z is a standard normal random variable then Z2 has the Chi-square distribution...

Show that if Z is a standard normal random variable then Z2 has the Chi-square distribution with one degree of freedom.

Let Y~ xî (i.e. Y follows a chi-squared with 1 degree of freedom), let Xn =...

Let Y~ xî (i.e. Y follows a chi-squared with 1 degree of freedom), let Xn = n-1/2 y - n1/2 (a) Show that Xn 4 X where X ~ N(0,2) (Hint: look up what is the mean and variance of a chi-squared distribution?) (b) n= 30. Find the exact P(Y > 43.8) using a chi-squared table. (c) Approximate P(Y > 43.8) for n = 30 using a normal approximation
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed,...

8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
7. Let Xn Xi++X2, where the Xi's are iid standard normal random variables (a) Show that Sn is a c...

7. Let Xn Xi++X2, where the Xi's are iid standard normal random variables (a) Show that Sn is a chi-square random variable with n de- grees of freedom. Hint: Show that X is chi-square with one degree of freedom, and then use Problem 6. (b) Find the pdf of (c) Show that T2 is a Rayleigh random variable. (d) Find the pdf for Ts. The random variable Ts is used to model the speed of molecules in a gas. It...

Let X1, X2, · · · be independent random variables, Xn ∼ U(−1/n, 1/n). Let X...

Let X1, X2, · · · be independent random variables, Xn ∼ U(−1/n, 1/n). Let X be a random variable with P(X = 0) = 1. (a) what is the CDF of Xn? (b) Does Xn converge to X in distribution? in probability?