session7

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 03/04/2014 for the course STAT 100A taught by Professor Wu during the Winter '10 term at UCLA.

Ask a homework question - tutors are online