A life insurance company sells a term insurance policy to a 21-year-old male that pays $100,000 if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $250 each year as payment for the insurance. The amount X that the company earns on this policy is $250 per year, less the $100,000 it must pay if the insured dies. Here is the distribution of X. Fill in the missing probability in the table and calculate the mean profit μx. 22 23 24 25 ≥26 Age 21 Profit -$99,750 -$99,500 -$99,250 -$99,000 -$98,750 $1250 Prob. .00183 .00186 .00189 .00191 .00193 Section 7.2 Second Day Rules for Means and Variances Law of Large Numbers   This is very important! This law says that As the number of observations increases, the sample mean approaches the idealized mean. In symbols, as n  , x    So, the more samples we get, the closer the mean is to what is “should” be. Rules for Means If X and Y are random variables, and a and b are fixed numbers, then a bX  a  b x  X Y   X  Y We will look at these individually in one minute… Simply Put  If you want to find the mean of the sum/difference of two random variables, you just add/subtract their means. – If the mean of X is 150 and the mean of Y is 1000, then the mean of X + Y = 150 + 1000.  x y   x   y The other rule abx  a  b x   This says that if we add a number, a, to each sample then we add a to the mean. Also, if we multiply every value in the sample by b then we have to multiply the mean by b. Rules for Means Demonstrated X = units sold for military division Probability 1000 3000 5000 10,000 0.1 0.3 0.4 0.2 Y = units sold for civilian division 300 500 750 Probability 0.4 0.5 0.1 If this company makes $2000 for each military unit sold and $3500 on each civilian unit sold, find the mean TOTAL profit. Rules for Variances If X is a random variable, and a and b are fixed numbers, then …  2 a bx b  2 2 x If we add the same number, a, to each item in the sample, it doesn’t change the variance. If we multiply each value by b then the standard deviation is multiplied by b, so the variance is multiplied by the square of b. When you have two variables…    We have to look at how the correlation between the two affect the variance of the sum of x and y The true correlation is called rho, ρ. The general rule for variances of random variables…  2 x y      2  x y  2 x y      2  x y 2 x 2 x 2 y 2 y What if they are independent?   If x and y are independent, they have no effect on one another, so…. ρ = 0 and therefore…  2 x y 2 x    2 y  2 x y    2 y 2 x Examples… If X and Y are independent random variables and…  X  5and  Y  2, find  X+Y. Find  X Y . Find  3 X 4Y . One more example:   Suppose Tom’s score in a round of golf is the random variable X and George’s score in a round of golf is the random variable Y. Assume their scores are independent. Find the average of their combined scores and the standard deviation of their combined scores if X  110; X  10 and Y  100; Y  8 More information…    One of the skills you need to learn from this section is combining two independent normal random variables and finding probabilities. Find the combined mean and standard deviation, and then work the problem as you would any normal curve probability (find the Z-score). In a fair game,   0 Example Go back to Tom and George’s golf game. What percent of the time would we expect Tom to win?  Tom:  = 110 =10  George:  = 100  = 8   (x-y) = 10  (x-y) = 12.8  Homework Chapter 6 # 36, 44, 56 Things to study for your test       How to test for independence Slides 16 and 17 from Chapter 6 Day 2 Calculating mean and std. deviation of a distribution Law of Large Numbers How mean and std. deviation are affected by addition and multiplication Review Sheet