Practice problems using various statistical methods
T Distribution arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. E(X)=0 ,VAR(X)=n/n-2, df=n
F Distribution arises frequently as the null distribution of a test statistic, most notably in the analysis of variance.E(X)=n/n-2
m,n is the degrees of freedom
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chi squared is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing or in construction of confidence intervals. E(X)=n, V(X)=2n, n is the degrees of freedom
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....
3, Let X, X2,X, be independent random variables such that Xi~N(?) a. Find the distribution of Y= a1X1+azX2+ i.(Hint: The MGF of Xi is Mx, (t) et+(1/2)t) + anXn +b where a, 0 for at least one b. Assume = 2 =n= u and of- a= (X-)/(0/n) ? Explain. a. What is the distribution of The Sqve o tubat num c. What is the distribution of [(X-4)/(0/Vm? Explain.
Let Xi, X2, , xn be independent Normal(μ, σ*) random variables. Let Yn = n Ση1Xi denote a sequence of random variables (a) Find E(%) and Var(%) for all n in terms of μ and σ2. (b) Find the PDF for Yn for all n c) Find the MGF for Y for all n
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y ~ N(5,9) (we use the notation N (?, ?. ) ). Let W 3X-2Y + 1. (a) Compute E(W) and Var(W) (b) It is known that the sum of independent normal distributions is n Compute P(W 6)
Given independent random variables, X and Y, with means and standard deviations as shown, find the mean and standard deviation of each of the variables in parts a to d. a) Xminus10 b) 0.5Y c) XplusY d) XminusY Mean SD X 60 13 Y 25 5 a) Find the mean and standard deviation for the random variable Xminus10. E(Xminus10)equals nothing SD(Xminus10)equals nothing (Type integers or decimals rounded to two decimal places as needed.)
5. The means, standard deviations, and covariance for random variables X, Y, and Z are given below. Ux= 3, uy = 5, uz = 7 Ox= 1, OY = 3, oz = 4 cov(X, Y) = 1, cov (X, Z) = 3, and cov (Y,Z) = -3 T= X-28 +3 Z var(T) = 16. For a random variable X with an unknown distribution. The mean of X is u = 22 and tting a randomly chosen value of X
Given independent random variables, X and Y, with both means and standard deviations shown below, find the mean and deviation of each of the variable in parts a through d. Mean Standard Deviation X 60 12 Y 12 4 a) 4X b) X + Y c) X - Y d) 3X - Y
der two independent random variables X and Y with the following 11. Consi means and standard deviations: = 60; ơv_ 15. (a) Find E(x + Y), Var(X + Y), E(X Y), Var(X - Y). (b) If x* and Y* are the standardized r.v.'s eorresponding to the r.v.'s X and Y, respectively, determine E(X* + Y*), E(X*-Y*), Var(X* Y*), Var(x* - Y*) der two independent random variables X and Y with the following 11. Consi means and standard deviations: = 60;...
1. Let Xi, X2,... be independent random variables each with the standard normal distribution, and for each n 2 0 let Sn-1 Xi. Use importance sampling to obtain good estimates for each of the following probabilities: (a) Pfmaxn<100 Sn> 10; and (b) Pímaxns100 Sn > 30) HINTS: The basic identity of importance sampling implies that d.P n100 where Po is the probability measure under which the random variables Xi, X2,... are independent normals with mean 0 amd variance 1. The...
Let X and Y independent random variables with standard normal distribution. Calculate = mln 772 272 , ly Answer: 210g (2)/n Why? = mln 772 272 , ly Answer: 210g (2)/n Why?
Let the independent normal random variables Y1,Y2, . . . ,Yn have the respective distributions N(μ, γ 2x2i ), i = 1, 2, . . . , n, where x1, x2, . . . , xn are known but not all the same and no one of which is equal to zero. Find the maximum likelihood estimators for μ and γ 2.