Practice Problems for Final Exam The final exam is cumulative so in addition to the problems below also go back to the practice problem sets for Midterms 1 and 2 1 Let the mgf of X be M t 2e2t e3t 3e 2t 6 a Find the mean and variance of X b Find P X 1 2 c Find the mean and variance of 3X 3 d Find the mean and variance of X 3 2 Scott and Dan have to complete 2 homeworks tonight one for the physics class and one for the Spanish class Scott and Dan work on their homeworks independently The time in minutes Scott and Dan spend on the hw for the physics class are X1 N 100 729 and X2 N 90 841 respectively And the time Scott and Dan spend on the hw for the Spanish class are Y1 N 40 121 and Y2 N 48 169 respectively a Find the probability that Scott will not finish his two homeworks within 3 hours b Find the probability that Scott will complete his two homeworks earlier than Dan by at least 20 minutes Hint Here if you add 20 min to Scott s time it s still less than Dan s time 3 X1 X2 X30 is a random sample from Exponential distribution with mean 10 a What is the exact distribution of cid 80 30 b What is the approximate distribution of cid 80 30 of the parameter s of this distribution value s of the parameter s of this distribution i 1 Xi Provide the name and the value s i 1 Xi Provide the name and the c What is the approximate distribution of X Provide the name and the value s of the parameter s of this distribution 4 Let X1 X2 X30 be a random sample from Normal distribution with mean 10 and variance 20 a What is the distribution of the sum of these random variables Provide the name and the value s of the parameter s of this distribution b What is the distribution of the sample average X of these random variables Provide the name and the value s of the parameter s of this distribution c What is the distribution of cS2 where S2 is the sample variance of these random variables and c is an appropriate constant What is the value of c Provide the name and the value s of the parameter s of this distribution 5 The yearly claim in dollars of a randomly chosen policyholder of MAICO an auto insurance company is known to have mean 750 and standard deviation 250 a Given that the average claim of 10 000 policyholders is above 745 what is the approximate probability that it is less than 753 b Find c such that the probability that the average claim is below c is 0 10 6 Suppose that 68 of New York City residents feel that the quality of life in NYC In a survey 1000 NYC residents were has become worse in the past few years asked whether the quality of life in NYC has become worse in the past few years Approximate the probability that the number of NYC residents that feel that the quality of life in NYC has become worse in the past few years is between 655 and 720 both inclusive State and check the appropriate conditions for the approximation 7 The mgf of X is M t 1 t 0 1 Find the variance of X Hint Recognize 5 1 10t the distribution of X based on this mgf 8 The mgf of X is M t et2 t Find the 99th percentile of X 9 Suppose X is normally distributed with mean 10 and variance 15 Find E e0 6X 10 X1 and X2 are iid random variables from the distribution with pmf f x x 1 6 x 0 1 2 a Find the expectation and standard deviation of Y 2X1 X2 b Find the mgf and the pmf of Y c Use the pmf of Y in b to find P Y 0 and E cid 112 Y 11 You are given the following information about random variables X Y V and W X is a random variable with the mgf MX t 1 Y is a random variable with the mgf MY s V is a random variable with the pdf fV v 1 W is a gamma random variable with mean 6 and variance 18 Corr X Y 0 5 Corr X V 0 5 Corr Y V 0 5 W is independent from the other 3 variables 2et 3 t 2 1 1 2 es 2 es 3 s ln 2 1 1 e x 3 2 6 v 6 a Find the mean and variance of X Y and V Hint Recognize the distributions b Find the covariances Cov X Y Cov X V and Cov Y V c Find E X 2W 2 d Find P X 2 W 2 e Find the mean and variance of X W f Find the mean and variance of X 3Y V g Find the covariance and correlation between X W and X 3Y V