7.2 Means and Variances of Random Variables

 Calculate the mean and standard deviation of random variables  Understand the law of large numbers

 You pick a three digit number in the lottery. If your number matches the states number, you win $500. What are your average winnings?  Probability Distribution  Probabilities are an idealized description of long- run proportions, so the mean of a probability distribution describes the average winnings in the long-run Outcome$500$0 Probability1/ /1000

 Suppose X is a discrete random variable whose distribution is  To find the mean of X, multiply each possible value by its probability, then add all the products: Value of Xx₁x₂x₃.... Probabilityp₁p₂p₃....

 Ex: The distribution of the count X of heads in four tosses of a balanced coin.  The expected value is: (0)(0.0625) + (1)(0.25)+….+(4)(0.0625)= 2 # of heads Prob

 The mean is the center of a symmetrical distribution.

 μ= 1(0.25)+2(0.32)+…+7(0.01)= 2.6 Inhabitants Probability

7.17 (0)(0.1)+(1)(0.15)+…+(4)(0.15)= Your payout is either $0 or $3 a) b) 0(0.75)+3(0.25)=$0.75 c) The casino makes $0.25 for every dollar you bet Payout$0$3 Probability

 7.19 If you choose a #, you could get: abc, acb, bac, bca, cab, cba Ex: 345, 354, 435, 453, 534, 543 0(0.994)+83.33(0.006)=$0.50 Payout$0$83.88 Probability

 Law of large numbers  Draw independent observations at random from any population with finite mean (μ).  Decide how accurately you would like to estimate the mean. As the number of observations drawn increases, the mean of the observed values eventually approaches the mean of the population as closely as you specified and then stays that close.  Describe this in your own words? When you increase your sample size, your sample mean gets closer to the true mean

 Reese’s example: ses3/ReesesPieces.html ses3/ReesesPieces.html  Ex: Use the average height of women to explain this. The mean is 64.5 in with a standard deviation of 2.5 in.

 Rule 1: If X is a random variable and a and b are fixed numbers, then:  Rule 2: If X and Y are (independent) random variables, then

 Ex: Military divisions  Civilian Division  Let x= # of military units sold y= # of civilian units sold  What is the mean number of military units sold? Civilian units sold? µ=1000(.1)+3000(.3)+…+10000(.2)=5000 units µ=300(.4)+500(.5)+750(.1)= 445 units Units Prob Units Prob

 If a profit of $2000 is made on each military unit sold and $3500 is made on each civilian unit, what is the total mean profit for units sold?

 The variance of X is:  So the standard deviation is: Value of Xx₁x₂x₃.... Probabilityp₁p₂p₃....

Then sum(L₃)

 Find the standard deviation of the military units sold?  Find the standard deviation of the civilian units sold?

 Rule 1:  Rule 2: ***we can’t add standard deviations, only variances!!!!***

 Ex: The payoff X of a $1 ticket in the Tri- State pick 3 game is $500 with probability 1/1000 and $0 the rest of the time. What is the variance of the total payoff if you buy $1 ticket on two different days? x$0$500 P(x)

 SAT math score X  SAT verbal score Y  What are the mean and standard deviation of the total score X + Y among students applying to this college? µ= =1215 σ= √(90²+100²)=134.54

 Tom’s score X:  George’s score Y:  Their scores vary independently. What is the mean difference between their scores? =10  What is the variance of the difference between their scores? 10²+8²=164  So the standard deviation is? √164=12.8

 What is the mean if I doubled everyone’s test score? 2(80)=160  What is the standard deviation? √ (2²*4²)=8

 What if I added 5 bonus points to everyone’s score. What is the new mean and standard deviation? μ=80+5=85 σ=4  What if I doubled everyone’s score and added 5 points. What is the new mean and standard deviation? μ=2(80)+5=165 σ= √ (2²*4²)=8