PDB_Stat_100_Lecture_15_Printable

PDB_Stat_100_Lecture_15_Printable - STA 100 Lecture 15 Paul...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
STA 100 Lecture 15 Paul Baines Department of Statistics University of California, Davis February 7th, 2011
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Admin for the Day I Please pick up Midterm+old homeworks I Important: Homework 3 Solutions posted – please read! I Homework 4 posted, due Wednesday by 5pm! I Hwk 4, Q2(i) – compare to the correct answer for Hwk3 Q2(k) I Hwk 4, Q3 – Weekly data (not monthly) I Project details coming soon. . . References for Today: Rosner, Ch 6.5-6.8, Ch 6.5 (7th Ed.) References for Wednesday: Rosner, Ch 6.5-6.8 (7th Ed.)
Background image of page 2
Recap: Linear Combinations We just covered linear transformations , where you add and multiply a single random variable by fixed numbers ( a and b ). Now we consider linear combinations of random variables. This is where you add multiple random variables together. . . These are different things! Linear transformations deal with transformations of a single variable (usually to change units), whereas linear combinations deal with what happens when you combine lots of random variables (usually to take a sum or average of them).
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Recap: Linear Combinations Let X 1 ,..., X n be any random variables, c 1 c n constants (just numbers). I The mean of a linear combination of any random variables: E [ c 1 X 1 + c 2 X 2 + ··· + c n X n ] = c 1 E [ X 1 ] + c 2 E [ X 2 ] + + c n E [ X n ] . I The variance of a linear combination of independent random variables: Var ( c 1 X 1 + c 2 X 2 + + c n X n ) = c 2 1 ( X 1 ) + c 2 2 ( X 2 ) + + c 2 n ( X n ) . I The SD of a linear combination of independent random variables: compute the variance and take a square root. Note: you cannot just add the standard deviations! I Linear combinations of normally distributed random variables are still normally distributed!
Background image of page 4
Recap: Linear Combinations Let X 1 ,..., X n be independent normally distributed random variables, c 1 c n constants (just numbers). I If we create a new random variable using the X ’s, say: Y = c 1 X 1 + c 2 X 2 + ··· + c n X n , then, using the properties on the previous slide we get: Y N ( μ Y 2 Y ) , where: μ Y = c 1 E [ X 1 ] + c 2 E [ X 2 ] + + c n E [ X n ] , σ 2 Y = c 2 1 Var ( X 1 ) + c 2 2 ( X 2 ) + + c 2 n ( X n ) .
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example Two independent random variables: X 1 N (0 , 1), X 2 N (0 , 1). I Lets consider Y = X 1 - X 2 . I So, compare to the general formula: Y = 1 · X 1 + ( - 1) · X 2 Y = c 1 · X 1 + c 2 · X 2 I So, c 1 = 1 and c 2 = - 1 . Plugging in: E [ Y ] = 1 · E [ X 1 ] + ( - 1) · E [ X 2 ] = 0 , Var ( Y ) = 1 2 · ( X 1 ) + ( - 1) 2 · ( X 2 ) ( Y ) = ( Y ) + ( Y ) = 1 + 1 = 2 , i.e., Y N (0 , 2) . I Even though we subtract X 1 and X 2 we add their variances!
Background image of page 6
Subtracting Normals Lots of X 1 ’s and X 2 ’s, Y ’s: x1 x2 x1-x2 1 -0.733730691 0.77965838 -1.51338907 2 -0.077958086 0.02981548 -0.10777357 3 -1.364323284 1.28076141 -2.64508470 4 -1.434933965 0.33073996 -1.76567393 5 1.179683727 1.27726560 -0.09758187 6 -0.008452536 -0.57704471 0.56859218 ...
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
X 1 and X 2 -6 -4 -2 0 2 4 6 0.0 0.1 0.2 0.3 0.4 Histogram of X_1 x1 -6 -4 -2 0 2 4 6 Histogram of X_2 x2
Background image of page 8
Y = X 1 - X 2 -4 -2 0 2 4 6 0.0 0.1 0.2 0.3 0.4 Histogram of Y = (X_1 - X_2) x1 - x2 Note: The spread is wider than for X 1 and X 2 !
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example Example:
Background image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 29

PDB_Stat_100_Lecture_15_Printable - STA 100 Lecture 15 Paul...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online