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assign1 - = Γ n m 2 Γ n 2)Γ m 2 ± n m n 2 x n 2-1 ...

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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.
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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 , σ 2 ), find the pdf of Y = | X | .
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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
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Unformatted text preview: ) = Γ( n + m 2 ) Γ( n 2 )Γ( m 2 ) · ± n m ! n/ 2 · x n/ 2-1 · ± 1 + n m x !-( n + m ) / 2 , x > ,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 β e-x-μ β , x > μ,β > (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 < t <-h Prove that P ( X ≥ a ) ≤ e-at M ( t ) , < t < h and P ( X ≤ a ) ≤ e-at M ( t ) ,-h < t < (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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