Show that y ni1 i xi where the i are scalars is

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Unformatted text preview: le when the variables are independent. Let Y= X1+ … +X20. Use M y to indicate moment generating function of y and use M x to indicate moment generating function of X, then € € 2 [STAT 100A/SANCHEZ TA SESSION] 7 20 M y = ∏ M x i = M x20 = e 20( e t −1) i= 1 Y follows the Poisson distribution with λ =20. e−20 20 y P (Y = y ) = y! € € € 4.- Let X1, X2,…..,Xn be independent normal random variables with mean µi and variance σ2i. Show that Y= ΣnI=1 αi Xi, where the αi are scalars, is normally distributed, and find its mean and variance. (Hint: use moment generating functions). First, we know that the moment generating function for normal distribution is: 22 M ( t ) = e µt +σ t / 2 Hence: t∑α X MY ( t ) = E (e tY ) = E (e i i ) = E (∏ e tα i X i ) = ∏ E (e tα i X i ) € n n n 2 = ∏ M X i (α i t ) = ∏ e ui α i t +σ i α i i= 1 22 t /2 t⋅ =e 2n ∑ ui α i + t2 ∑σ i2α i2 i =1 i =1 i= 1 Comparing the moment generation function of Y with the moment generation of normal distribution, we know that Y is also normal distributed with n € Mean = ∑ uiα i i= 1 n Variance = ∑σ i2α i2 i= 1 Practicing discrete bivariate distributions € 5. Suppose that 15 percent of the families in a certain community have no children, 20 percent h...
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This note was uploaded on 03/04/2014 for the course STAT 100A taught by Professor Wu during the Winter '10 term at UCLA.

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