Sums of Independent Random Variables Lincar combinations a 1 x 1 a 2 x 2 cdots a k x k i 1 k a i x i Linear Combinations of Independent Normal Random Variables Proposition Let x 1 , x 2 , dots, x k be independent and x i N ( i , i 2 ) , i 1 , 2 , dots, k Then a 1 x 1 a 2 x 2 cdots a k x k i 1 k a i x i , N ( i 1 k a i i , i 1 k a i 2 i 2 ) Sums of Independent Random Variables ( Not Normal ) Proposition If x 1 , x 2 , dots, x k are independent and x i B i n o m ( n i , p ) , i 1 , 2 , , k Then i 1 k x i B i n o m ( i 1 k n i , p ) If x 1 , x 2 , dots, x k are independent and x i N e g B i n o m ( r i , p ) , i 1 , 2 , dots, k Then i 1 k x i N e g B i n o m ( i 1 k r i , p ) If x 1 , x 2 , dots, x k are independent and x i P o i s ( i ) , i 1 , 2 , dots, k Then i 1 k x i P o i s ( i 1 k i ) If x 1 , x 2 , dots, x k are independent and x i ( i , ) , i 1 , 2 , dots, k Then Example Suppose the useful life ( in years ) of a refrigerator is normally distributed The information for the useful life of three most common brands is summarized in the table table brand , mean,standard deviation , 1 , 1 0 , 3 , 2 , 9 5 , 2 , 3 , 1 1 , 4 The useful lives of refrigerators are independent Suppose a refrigerator is randomly selected from each brand Find the probability that the total useful life of the refrigerators from the first two brands will exceed 1 9 times the useful life of the refrigerator from the 3 rd brand Practice 1 Let x have a normal distribution with mean 1 0 and variance 5 Let Y have a normal distribution with mean 1 2 and variance 5 Suppose x and Y are independent a ) Find the probability that x Y is greater than 2 5 b ) Find the probability that x Y is greater than 2 Practice 2 Let x be Binomial with parameters 1 0 and 0 5 and Y be binomial with parameters 1 2 and 0 5 Suppose these variables are independent Find the probability that x Y is equal to 1 6 SOLVE THE PROBLEM USING THE NOTES FROM PAGE 1 Show all images Show all images Show all images done loading

Question

Sums of Independent Random Variables Lincar combinations  a 1 x 1 a 2 x 2 cdots a k x k   i   1 k a i x i Linear Combinations of Independent Normal Random Variables Proposition  Let x 1 , x 2 , dots, x k be independent and x i N ( i , i 2 ) , i   1 , 2 , dots, k   Then a 1 x 1 a 2 x 2 cdots a k x k   i   1 k a i x i ,   N ( i   1 k a i i , i   1 k a i 2 i 2 ) Sums of Independent Random Variables ( Not Normal ) Proposition  If x 1 , x 2 , dots, x k are independent and x i B i n o m ( n i , p ) , i   1 , 2 ,   , k   Then i   1 k x i B i n o m ( i   1 k n i , p ) If x 1 , x 2 , dots, x k are independent and x i N e g B i n o m ( r i , p ) , i   1 , 2 , dots, k   Then i   1 k x i N e g B i n o m ( i   1 k r i , p ) If x 1 , x 2 , dots, x k are independent and x i P o i s ( i ) , i   1 , 2 , dots, k   Then i   1 k x i P o i s ( i   1 k i ) If x 1 , x 2 , dots, x k are independent and x i ( i , ) , i   1 , 2 , dots, k   Then Example  Suppose the useful life ( in years ) of a refrigerator is normally distributed  The information for the useful life of three most common brands is summarized in the table     table     brand , mean,standard deviation   ,   1 , 1 0 , 3   ,   2 , 9   5 , 2   ,   3 , 1 1 , 4     The useful lives of refrigerators are independent  Suppose a refrigerator is randomly selected from each brand  Find the probability that the total useful life of the refrigerators from the first two brands will exceed 1   9 times the useful life of the refrigerator from the 3 rd brand  Practice 1   Let x have a normal distribution with mean 1 0 and variance 5   Let Y have a normal distribution with mean 1 2 and variance 5   Suppose x and Y are independent  a ) Find the probability that x Y is greater than 2 5   b ) Find the probability that x   Y is greater than 2   Practice 2   Let x be Binomial with parameters 1 0 and 0   5 and Y be binomial with parameters 1 2 and 0   5   Suppose these variables are independent  Find the probability that x Y is equal to 1 6   SOLVE THE PROBLEM USING THE NOTES FROM PAGE 1 Show all images Show all images Show all images done loading

HelloExperts · Accepted Answer

The Answer is in the image, click to view...

Sums of Independent Random Variables Lincar combinations: a 1 x 1 a 2 x 2 cdots a k x k = i = 1 k

Step by Step Solution

1 Understanding Responsive Design Principles

Related Time Series Analysis Questions