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PROBLEM 8: It is easy to show with mgf's that the sum of independent chi-squared random...
Solve b 3.6.14. Let Xi, X2, and Xs be three independent chi-square variables with rı, r2, and T3 degrees of freedom, res...
Solve b
3.6.14. Let Xi, X2, and Xs be three independent chi-square variables with rı, r2, and T3 degrees of freedom, respectively. (a) Show that Yı = X1/X2 and ½ = X1 + X2 are independent and that ½ is x 2) (b) Deduce that and KJr2 (X1+X2)/(n+r2) are independent F-variables.
3.6.14. Let Xi, X2, and Xs be three independent chi-square variables with rı, r2, and T3 degrees of freedom, respectively. (a) Show that Yı = X1/X2 and ½ =...
Practice problems using various statistical methods If n independent random variables X have normal distributions with means μ and the standard deviations σ , then determine the distribution of a. I....
Practice problems using various statistical methods
If n independent random variables X have normal distributions with means μ and the standard deviations σ , then determine the distribution of a. I. X-E(X) var(X) C. 2. If n independent random variables Xi have normal distributions with means μί and the standard deviations σί, then determine the distribution of a. b. Y -a1X1 + a2X2+ + anXn (ai constant) X-E(X) Vvar(X) 3. What is CLT? Proof briefly? What are t-, Chi-squared- and...
Problem 3: Understanding the chi-squared test of independence of categorical random variables The problem has two...
Problem 3: Understanding the chi-squared test of independence of categorical random variables The problem has two sub-problems that should help understand the chi-squared test of independence. a) For the chi-squared test of independence between two categorical variables, we use the fact that the MLE for the probability that a multinomial trial is of category i is ni /n, where ni is the number of trials that fall into category i, and n is the total number of trials. Prove the...
21 (1 point) If Y X and every Xi is i.i.d with a chi-squared distribution with...
21 (1 point) If Y X and every Xi is i.i.d with a chi-squared distribution with 14 degrees of freedom, find the MGF of Y М()- What is the distribution of Y? Select all that apply. There may be more than one correct answer. А. gатта(а - 147, 8 В. датта(а — 2, В — 294) с. датта(а — 1,8— 1/147) D. gamma(a 1, B 1/294) E. chi squared(df 294) F. еzрoпential(A — 204) G. exponential(A 147) H. chi squared(df...
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)
(1 point) In Unit 3, I claimed that the sum of independent, identically distributed exponential r...
(1 point) In Unit 3, I claimed that the sum of independent, identically distributed exponential random variables is a gamma random variable. Now that we know about moment generating functions, we can prove it. Let X be exponential with mean A 4. The density is 4 a) Find the moment generating function of X, and evaluate at t 3.9 The mgf of a gamma is more tedious to find, so l'll give it to you here. Let W Gamma(n, A...
In this problem we show directly that the sum of independent Poisson random variables is Poisson....
In this problem we show directly that the sum of independent Poisson random variables is Poisson. Let J and K be independent Poisson random variables with expected values α and respectively. Show that Ņ J+K is a Poisson random variable with expected value α+β Hint: Show that 72 Pk (m)P, (n-m and then simplify the summation be extracting the sum of a binomial PMF over al possible values.
In this problem we show directly that the sum of independent Poisson...
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...
explan the answer . Suppose that Xi, X2,.... Xn are independent random variables. Assume that E[A]-:...
explan the answer . Suppose that Xi, X2,.... Xn are independent random variables. Assume that E[A]-: μί ald Var(Xi)-σ? where i-| , 2, , n. If ai, aam., an are constants. (i) Write down expression for (i) E{E:-aiX.) and (ii) Var(Σ-lai%). (i) Rewrite the expression if X,'s are not independent.
explan the answer 1l. Suppose that X1, X2,... Xn are independent random variables. Assume that ElXi]...
explan the answer
1l. Suppose that X1, X2,... Xn are independent random variables. Assume that ElXi] /4 and Var(X )-σ, where i 1, 2, . .., n. If ai , aam. , an are constants. 1,a2, , an are constan (i) Write down expression for (i) E{Σ,i ai Xi) and (ii) Var(Li la(Xi). (i) Rewrite the expression if X,'s are not independent.
Solve b
3.6.14. Let Xi, X2, and Xs be three independent chi-square variables with rı, r2, and T3 degrees of freedom, respectively. (a) Show that Yı = X1/X2 and ½ = X1 + X2 are independent and that ½ is x 2) (b) Deduce that and KJr2 (X1+X2)/(n+r2) are independent F-variables.
3.6.14. Let Xi, X2, and Xs be three independent chi-square variables with rı, r2, and T3 degrees of freedom, respectively. (a) Show that Yı = X1/X2 and ½ =...
Practice problems using various statistical methods
If n independent random variables X have normal distributions with means μ and the standard deviations σ , then determine the distribution of a. I. X-E(X) var(X) C. 2. If n independent random variables Xi have normal distributions with means μί and the standard deviations σί, then determine the distribution of a. b. Y -a1X1 + a2X2+ + anXn (ai constant) X-E(X) Vvar(X) 3. What is CLT? Proof briefly? What are t-, Chi-squared- and...
21 (1 point) If Y X and every Xi is i.i.d with a chi-squared distribution with 14 degrees of freedom, find the MGF of Y М()- What is the distribution of Y? Select all that apply. There may be more than one correct answer. А. gатта(а - 147, 8 В. датта(а — 2, В — 294) с. датта(а — 1,8— 1/147) D. gamma(a 1, B 1/294) E. chi squared(df 294) F. еzрoпential(A — 204) G. exponential(A 147) H. chi squared(df...
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)
(1 point) In Unit 3, I claimed that the sum of independent, identically distributed exponential random variables is a gamma random variable. Now that we know about moment generating functions, we can prove it. Let X be exponential with mean A 4. The density is 4 a) Find the moment generating function of X, and evaluate at t 3.9 The mgf of a gamma is more tedious to find, so l'll give it to you here. Let W Gamma(n, A...
In this problem we show directly that the sum of independent Poisson random variables is Poisson. Let J and K be independent Poisson random variables with expected values α and respectively. Show that Ņ J+K is a Poisson random variable with expected value α+β Hint: Show that 72 Pk (m)P, (n-m and then simplify the summation be extracting the sum of a binomial PMF over al possible values.
In this problem we show directly that the sum of independent Poisson...
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...
explan the answer . Suppose that Xi, X2,.... Xn are independent random variables. Assume that E[A]-: μί ald Var(Xi)-σ? where i-| , 2, , n. If ai, aam., an are constants. (i) Write down expression for (i) E{E:-aiX.) and (ii) Var(Σ-lai%). (i) Rewrite the expression if X,'s are not independent.
explan the answer
1l. Suppose that X1, X2,... Xn are independent random variables. Assume that ElXi] /4 and Var(X )-σ, where i 1, 2, . .., n. If ai , aam. , an are constants. 1,a2, , an are constan (i) Write down expression for (i) E{Σ,i ai Xi) and (ii) Var(Li la(Xi). (i) Rewrite the expression if X,'s are not independent.
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