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# assign_2 - e-y/β y> respectively(1 Find P X> Y(2 Find...

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1 Assignment #2 - STAT 330 Due in class: Tuesday June 14 Important Note: You need to print out this page as the cover page for your assignment. LAST NAME: FIRST NAME: ID. NO.: QUESTION 1. /8 QUESTION 2. /4 QUESTION 3. /8 QUESTION 4. /6 QUESTION 5. /10 QUESTION 6. /4 TOTAL: /40

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2 1. Suppose X and Y are discrete random variables with joint pmf f ( x, y ) = e - 2 x !( y - x )! , x = 0 , 1 , ..., y ; y = 0 , 1 , ..., (a). Are X and Y independent? (b). Find the marginal pmf’s of X and Y . (c). Find the conditional pmf of Y given X = x .
3 2. Suppose the joint cdf of X and Y is F ( x, y ) = (1 - e - αx )(1 - e - βy ) , x 0 , y 0 , α > 0 , β > 0 . (a). Find the joint pdf. (b). find the marginal pdf of X . (c). Find the joint mgf M ( t 1 , t 2 ). For what values of t 1 and t 2 , does M ( t 1 , t 2 ) exist?

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4 3. Suppose that X and Y are independent random variables. (a). If the marginal distribution of both X and Y is the standard normal distribution, find P ( X 2 + Y 2 1). (b). Suppose the marginal distributions of X and Y exponentially distributed with parameters α and β , respectively. That is, the marginal pdf of X ans Y are f ( x ) = 1 α e - x/α , x > 0 , and f ( y

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Unformatted text preview: e-y/β , y > , respectively. (1). Find P ( X > Y ). (2). Find P ( X > 2 Y ). 5 4. For each of the following questions, ﬁnd E ( X ) and V ar ( X ). (a). Suppose X | Y = y ∼ POI( y ), and Y ∼ GAM( α,β ) with the pdf f ( y ) = 1 Γ( α ) β α y α-1 e-y/β , for y > . (b). Suppose X | Y = y ∼ BIN( y,p ) and Y ∼ POI( λ ). (c). Suppose X | Y = y ∼ BIN( y + 1 ,p 1 ) and Y ∼ BIN( n,p 2 ). 6 5. Let X and Y have the joint density f ( x,y ) = e-y , ≤ x ≤ y (a) Find COV ( X,Y ) and correlation of X and Y (b) Find E ( X | Y = y ) and E ( Y | X = x ) (c) Find the density functions of the random variables E ( X | Y ) and E ( Y | X ) 7 6. Suppose X and Y have the joint pdf f ( x,y ) = e-x-y , x > , y > , and zero otherwise. (a). Find P ( X + Y < 1). (b). Let U = X + Y , and V = X . Find the joint pdf of U and V ....
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