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Unformatted text preview: ve a multivariate normal distribution with mean vector 0
and variance-covariance matrix 1 0 0 = 0 2 1 .
0 1 2 Find P ( X 1 > X 2 + X 3 + 2 ).
Hint: Find the vector a so that a X = X 1 – X 2 – X 3 and make use of Theorem 3.5.1.
Suppose X has a N n ( , ) distribution. Let Y = A X + b, where A
is an m n matrix and b Rm. Then Y has a N m ( A + b, A A' ) distribution. Theorem 3.5.1. 12. If the price of gasoline goes up, the prices of the other goods tend to go up too.
Suppose the price of a gallon of gas ( X ) and the price of a gallon of milk ( Y )
at Anytown follow a bivariate normal distribution with X = $3.45,
a) X = $0.15, Y = $3.27, Y = $0.09, = 0.60. Find the probability that the price of a gallon of milk is below $3.45, if the price
of a gallon of gas is $3.70. That is, find P ( Y < 3.45 | X = 3.70 ). b) Find the probability that a gallon of gas costs more than a gallon of milk. That is,
find P ( X > Y ). c) Alex buys 5 gallons of gas and 2 gallons of milk. What is the probability that he
paid more than $25? That is, find P ( 5 X + 2 Y > 25 ). ____________________________________________________________________________
____________________________________________________________________________ If you are registered for 4 credit hours:
13. ( to be handed in on a separate sheet ) Let X and Y have the joint probability density function f X, Y ( x, y ) = C e – x2 , 0 < y < x < ∞, zero elsewhere. a) What must the value of C be so that f X, Y ( x, y ) is a valid joint p.d.f.? b) Find the marginal probability density function for X, f X ( x ). c) Find the expression for E ( X k ), k > – 2.
Hint: Def. x u x 1 e u du , x > 0. 0 d) Find the expression for the marginal probability density function for Y, f Y ( y ),
in terms of the c.d.f. of standard normal distribution, ( )....
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