# STAT 1430 Lecture Notes - Lecture 14: Random Variable, Gm Family 0 Engine, Weighted Arithmetic Mean

Thursday, February 21, 2019

STAT1430.01—Lecture14—Random Variables

-The mean of a discrete random variable

•A weighted average of the possible outcomes; weights are the probabilities

-Possible outcomes: x1, x2, … xk Call them x

-Probabilities: p1, p2, … pk Call them P(x)

•Notation: !

•Formula: !

-The mean of a discrete random variable

•Average of all possible values in whole population and The overall expected value of x

•!

-!=average of sample

-! (constant)= average of population

-Discrete example: Family size (X)

•Mean family size is:

-!=1*0+2*0.42+3*0.23+4*0.21+5*0.09+6*0.03+7*0.02=3.14

•Interpret this result:

-average family size for the entire population

-Rules of Means #1—Linear Transformations

•Linear transformation:

-multiply x by constant and/or add a constant.

•Example:

-10% salary raise + $500 bonus

-Y=1.10x + 500

•Rule:

μx

μx=∑xp(x)

¯

X≠μX

¯

X

μX

Numbers

1

2

3

4

5

6

7

Family

Probability

0

0.42

0.23

0.21

0.09

0.03

0.02

μx=∑

x

xp(x)

1

## Document Summary

The mean of a discrete random variable: a weighted average of the possible outcomes; weights are the probabilities. Possible outcomes: x1, x2, xk call them x. Probabilities: p1, p2, pk call them p(x: notation: , formula: ! The mean of a discrete random variable: average of all possible values in whole population and the overall expected value of x. Average family size for the entire population. Rules of means #1 linear transformations: linear transformation: Multiply x by constant and/or add a constant: example: A x+b = y = a x + b. Y = 1. 10 x + 500: note: this rule holds for any x, y. Example: cancun temperatures: the mean temperature in cancun is 28. 5 degrees centigrade ( c). What is the mean temperature in fahrenheit (f): f=9/5 c + 32. Example: 0 degrees c = 32 degrees f: mean of f = ! Rule #2 of means sums and differences: rule: 2 random variables x and y.