Class13-probability and random variables wrapup
.pdf
School
CUNY Hunter College**We aren't endorsed by this school
Course
ECO 221
Subject
Economics
Date
Dec 18, 2024
Pages
17
Uploaded by CommodoreStar20599
ECO 221Economic StatisticsProfessor Sangita VyasClass 13
the mean of RV X!!= #"$"+ ##$#+ … + #$$$or!!= Σ%#%$%
Rules for means of random variablesRule 1. If Xis a random variable and aand bare fixed numbers, then !!"#$= # + %!$.Rule 2.If Xand Yare random variables, then !$"%= !$+ !%.
the variance of RV X)&#= (#"−!!)#$"+ (##−!!)#$#+ … (#$−!!)#$$or)&#= Σ%(#%− !!)#$%
independence of random variablestwo random variables Xand Yare independent if knowing the outcome of Xdoesn’t tell us anything about the outcome of Y.
rules for variances of random variablesRule 1. If Xis a random variable and aand bare fixed numbers, then &!"#$&= %&&$&.Rule 2.If Xand Yare independent random variables, then &$"%&= &$&+ &%&and &$'%&= &$&+ &%&.Rule 3.If Xand Yhave correlation ', and are therefore not independent, then &$"%&= &$&+ &%&+2'&(&)and &$'%&= &$&+ &%&− 2'&(&).
Example - lotteryThere is a lottery in which you choose a three-digit number, 000 to 999. If the lottery organizer chooses your number then you win $500. What is the standard deviation of winnings from buying a lottery ticket?What are your net winnings if the price of ticket is $1? What is the standard deviation of net winnings?Suppose you buy 1 lottery ticket on 2 separate days. The drawings are held each day. What is the combined payoff? What is the standard deviation of the combined payoff?
Example SAT scoresYou randomly select a student from the population of students who took the SAT last year and you observe their Math score and their Verbal score.The means and standard deviations are:Math score mean: 519, st.dev: 115Verbal score mean: 507, st.dev: 111What is the mean and standard deviation of the total score? (Top Hat)
Example SAT scoresYou randomly select a student from the population of students who took the SAT last year and you observe their Math score and their Verbal score.The means and standard deviations are:Math score mean: 519, st.dev: 115Verbal score mean: 507, st.dev: 111Correlation between math and verbal scores is 0.71.What is the standard deviation of the total score? (Top Hat)
We learn about all of this as if we know the probability distribution of a RV, and therefore we know !. But in most cases, we actually don’t know the probability distribution of the random variable and therefore we can’t estimate !. We use random samples to estimate !.
Law of Large Numbers
some notationsample statisticpopulation parameterWhat is it?!"#means$Standard deviationr%correlation
Law of large numbersIn a population, the variable you are interested in has a mean, !.Draw independent observations at random from a population, and compute the mean ̅,of the sample. Draw a bigger sample, and compute ̅,again. Draw a bigger sample, and compute ̅,again. The law of large numbers says that ̅,will approach !as the sample size increases.
Why is the LLN important?We use random samples to estimate !. The LLN is helpful because it tells us that if we make our sample bigger and bigger and bigger, we will get closer to the true probability distribution in the population, and closer to !.Intuitively, it makes sense that when we sample the entire population, we get the true probability distribution and !.
Law of large numbershttps://demonstrations.wolfram.com/IllustratingTheLawOfLargeNumbers/
Tricky homework problems•Number 9•Number 8•Number 6