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Let Y~ xî (i.e. Y follows a chi-squared with 1 degree of freedom), let Xn =...
Let X be chi-squared with k = 5 degrees of freedom. (a) What is P[X >...
Let X be chi-squared with k = 5 degrees of freedom. (a) What is P[X > 1]? (This exact calculation requires some serious integration) (b) Find P[X > 1] with the normal approximation, i. e., approximate that probability assuming X ~ N(mu, sigma squared), where mu and sigma squared are the mean and variance of a Gamma(5/2, 1/2).
Let Yn be a chi square random variable with n degrees of freedom, and let Xn =...
Let Yn be a chi square random variable with n degrees of freedom, and let Xn = Yn / n2. Find the limiting distribution of Xn.
9. Show that Y-Σ|-1 z? has a Chi-squared distribution with n degrees of freedom, where Z's...
9. Show that Y-Σ|-1 z? has a Chi-squared distribution with n degrees of freedom, where Z's are iid rvs from a standard normal distribution. (Prove it mathematically)
10) Suppose that X follows a chi-square distribution with m degrees of freedom and S=X+Y. Given...
10) Suppose that X follows a chi-square distribution with m degrees of freedom and S=X+Y. Given that S follows a chi-square distribution with m+n degrees of freedom, and X and Y are independent, show that y follows a chi-square distribution with n degrees of freedom.
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, ..., Xn be a random sample from the N(u, 02) distribution. Derive a...
Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)
(a) Let X, have a chi-squared distribution with parameter V, and let X, be independent of...
(a) Let X, have a chi-squared distribution with parameter V, and let X, be independent of X, and have a chi-squared distribution with parameter vz. Show that X, + X, has a chi-squared distribution with parameter v, + V Let Y = X1 + Xy. Identify the correct expression for Fly). Fyly) = (f1 +49 (0) BM={{1*(**) 2*3)**- _jei OFW - 1 -{{49)..:@) ib) dx1 FY) = -xq12 dx + -*2/20 1²ax 2 1/2 O Fy(y) :{"(****): 19). x 22...
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e.,...
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. Show that SZ N(0,1) 5. Let Z N(0, 1) and X = Z2. This distribution is called chi-square with degree of freedom. Calculate P(1 < X < 4) one
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...
3. Let X be normal random variable and Y be a Chi-square random variable with df...
3. Let X be normal random variable and Y be a Chi-square random variable with df degrees of freedom then the ratio follows (note that this is the reason we use a common test when We don't know for certain the true value of the variance): a) A x?distribution b) A normal distribution c) An F distribution d) At distribution.
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1),...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
9. Show that Y-Σ|-1 z? has a Chi-squared distribution with n degrees of freedom, where Z's are iid rvs from a standard normal distribution. (Prove it mathematically)
10) Suppose that X follows a chi-square distribution with m degrees of freedom and S=X+Y. Given that S follows a chi-square distribution with m+n degrees of freedom, and X and Y are independent, show that y follows a chi-square distribution with n degrees of freedom.
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, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)
(a) Let X, have a chi-squared distribution with parameter V, and let X, be independent of X, and have a chi-squared distribution with parameter vz. Show that X, + X, has a chi-squared distribution with parameter v, + V Let Y = X1 + Xy. Identify the correct expression for Fly). Fyly) = (f1 +49 (0) BM={{1*(**) 2*3)**- _jei OFW - 1 -{{49)..:@) ib) dx1 FY) = -xq12 dx + -*2/20 1²ax 2 1/2 O Fy(y) :{"(****): 19). x 22...
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. Show that SZ N(0,1) 5. Let Z N(0, 1) and X = Z2. This distribution is called chi-square with degree of freedom. Calculate P(1 < X < 4) one
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...
3. Let X be normal random variable and Y be a Chi-square random variable with df degrees of freedom then the ratio follows (note that this is the reason we use a common test when We don't know for certain the true value of the variance): a) A x?distribution b) A normal distribution c) An F distribution d) At distribution.
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
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