Download Lecture 9 - Simon Fraser University

Statistics 270 - Lecture 9
• Last day/Today: Discrete probability distributions
• Assignment 3: Chapter 2: 44, 50, 60, 68, 74, 86, 110
Mean and Variance for Discrete Random
Variables
• Suppose have 1000 people in a population (500 male and 500
female) and average age of the males is 26 and average age of
females is 24
• What is the mean age in the population?
• Suppose have 1000 people in a population (900 male and 100
female) and average age of males is 26 and average age of females
is 24
• What is the mean age in the population?
• Mean must consider chance of each outcome
• Mean is not necessarily one of the possible outcomes
• Is a weighted average of the outcomes
Mean of a Discrete Random Variable
• The mean (or expected value) of a discrete R.V., X, is denoted E(X)
k
E ( X )   p ( xi ) xi   X
i 1
• Can be viewed as a long run average
Mean of a Bernoulli Random Variable
• p(x)=
• E(X)=
Example
•
k / x 2 for x  1,2,3,...
p ( x)  
0 otherwise

• E(X)=
Expected Value of a Function of a Random Variable
• Let h(X) be a function of a random variable, X
•The expected value of h(X) is
•E(h(X))=
Example
•
Let X be the rv denoting the January noon-time temperature at the
Vancouver International Airport
•
If the mean temperature is 5 oC, what is the mean temperature in
Fahrenheit?
Expectation Under Linear Transformations
•
E(aX+b)=
Variance of a Discrete Random Variable
• Variance is the mean squared deviation from the mean
• Squared deviation from mean
Variance of a Discrete Random Variable
• Variance of a discrete R.V. weights the squared deviations from the
mean by the probabilities
k
  Var ( X )   ( xi   ) 2 p( xi )
2
i 1
• The standard deviation is

k
 (x
i 1
i
  ) 2 p ( xi )
Variance of a Discrete Random Variable
• Alternate Formula for Variance:
Example
•
Let X be the rv denoting the January noon-time temperature at the
Vancouver International Airport
•
If the mean temperature is 5 oC and the variance is 3 (oC)2 , what
is the variance of the temperature in Fahrenheit?
Example
• Probability distribution for number people in a randomly selected
household
X=# people
p(xi)
1
2
3
4
5
6
7
0.25 0.32 0.17 0.15 0.07 0.03
Example
• Compute mean and variance of number of people in a household
Example (true story)
• People use expectation in real life
• Parking at Simon Fraser University was $9.00 per day
• Fine for parking illegally is $10.00
• When parking illegally, get caught roughly half the time
• Should you pay the $9.00 or risk getting caught?
Example
• In a game, I bet X dollars
• With probability p, I win Y dollars
• What should X be for the game to be fair?
w
p(w)
-X
1-p
Y
p
• Rules for Variance: