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# assign1 - ) = ( n + m 2 ) ( n 2 )( m 2 ) n m ! n/ 2 x n/...

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1 Assignment #1 - STAT 330 Due in class: Thursday Jan. 21 Important Note: You need to print out this page as the cover page for your assignment. LAST NAME: FIRST NAME: ID. NO.: QUESTION 1. QUESTION 2. QUESTION 3. QUESTION 4. QUESTION 5. QUESTION 6. TOTAL:

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2 1. Let f ( x ) = (1 + αx ) / 2 , - 1 x 1 0 , o.w. , where - 1 α 1. (a). Show that f is a density. (b). Find the corresponding c.d.f.
3 2. Let F ( x ) = 0 , x < 0 1 - e - αx β , x 0 where α > 0 ,β > 0 . (a). Show that F is a c.d.f. (b). Find the corresponding density.

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4 3. If X N (0 2 ), ﬁnd the pdf of Y = | X | .
5 4. Let X be a random variable having a t distribution with degrees of freedom n . Show that Y = X 2 follows an F distribution. Identify its degrees of freedom. Note the following deﬁnitions: (a). If X is a random variable having the pdf f ( x ) = Γ( n +1 2 ) Γ( n 2 )Γ( 1 2 ) · 1 n · ± 1 + x 2 n ! - ( n +1) / 2 , -∞ < x < , n = 1 , 2 ,... then X is called following a t distribution with degrees of freedom n , and denoted by X t ( n ). (b). If X is a random variable having the pdf f ( x

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Unformatted text preview: ) = ( n + m 2 ) ( n 2 )( m 2 ) n m ! n/ 2 x n/ 2-1 1 + n m x !-( n + m ) / 2 , x &gt; ,n,m = 1 , 2 ,... then X is called following an F distribution with degrees of freedom n and m , and denoted by X F ( n,m ). 6 5. Suppose X is a random variable with pdf f ( x ) = 1 e-x- , x &gt; , &gt; (a). Find m.g.f M ( t ) of X . For what values of t does M ( t ) exist ? (b). Find E ( X k ) for a positive integer k . 7 6. (a). Let X be a random varaible with mgf M ( t ) ,-h &lt; t &lt;-h Prove that P ( X a ) e-at M ( t ) , &lt; t &lt; h and P ( X a ) e-at M ( t ) ,-h &lt; t &lt; (b). Suppose the mgf of X exists for all real values of t and is given by M ( t ) = e t-e-t 2 t , t 6 = 0 M (0) = 1 Show that P ( X 1) = 0 and P ( X -1) = 0...
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## This note was uploaded on 06/21/2010 for the course STAT 5023 taught by Professor Yi,graceyun during the Spring '10 term at Waterloo.

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assign1 - ) = ( n + m 2 ) ( n 2 )( m 2 ) n m ! n/ 2 x n/...

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