Statistics 270 - Lecture 9. Last day/Today: Discrete probability distributions Assignment 3: Chapter 2: 44, 50, 60, 68, 74, 86, 110.

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Presentation transcript:

Statistics 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) Can be viewed as a long run average

Mean of a Bernoulli Random Variable p(x)= E(X)=

Example 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 o C, 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 The standard deviation is

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 o C and the variance is 3 ( o C) 2, what is the variance of the temperature in Fahrenheit?

Example Probability distribution for number people in a randomly selected household

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?

Rules for Variance: