a1su112131 - 1(b(5 marks Let X 1,X 2 be i.i.d r.v.’s with...

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MATH 2131 3.00 MS2 Assignment 1 Total marks = 40 Question 1: Let ( X,Y ) have the joint probability density function f X,Y ( x,y ) = ( cx/y 2 , y - 1 < x < y, 1 < y < 2 , 0 , otherwise. (a) (4 marks). Find c such that f X,Y ( x,y ) is a joint probability density function. (b) (4 marks). Find f X ( x ). (c) (4 marks). Find P ( X > 1 ,Y < 2). Question 2: A university club consists of 4 men and 6 women. Two members are chosen at random without replacement. Let X = ( 0 , if first person chosen is male, 1 , if first person chosen is female, Y = ( 0 , if second person chosen is male, 1 , if second person chosen is female. (a) (4 marks). Find p X,Y ( x,y ). (b) (2 marks). Find p Y ( y ). Question 3: (a) (5 marks). Two fair dice are rolled. Let r.v.’s X and Y denote the number of dots showing on the first and second dice, respectively. Show that X and Y are independent if their joint probability distribution function is given by p X,Y ( x,y ) = ( 1 / 36 , x = 1 , 2 ,..., 6; y = 1 , 2 ,..., 6 , 0 , otherwise.
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Unformatted text preview: 1 (b) (5 marks). Let X 1 ,X 2 be i.i.d. r.v.’s with common probability density function f ( x ) = ( e-x , x > , , otherwise. Find P ( X 1 + X 2 > 2) . Question 4: Let ( X,Y ) have the joint probability distribution function f X,Y ( x,y ) = ( (6-x-y ) / 8 , < x < 2 , 2 < y < 4 , , otherwise. (a) (5 marks). Find f Y | X ( y | x ). (b) (3 marks). Find P (2 < Y < 3 | X = 1) . Question 5: (4 marks). The opening market prices per bushel for corn and wheat are known to independently each follow the probability distribution function f ( x ) = ( e-( x-5) , x ≥ 5 . , otherwise. An investor decides she will buy whichever commodity has the lower price. Find the probability density function for the price per bushel that the investor will have to pay. 2...
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a1su112131 - 1(b(5 marks Let X 1,X 2 be i.i.d r.v.’s with...

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