1. 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 μ x. Age2122232425≥26 Profit-$99,750-$99,500-$99,250-$99,000-$98,750$1250 Prob..00183.00186.00189.00191.00193

2. Section 6.2 Rules for Means and Variances

3. Law of Large Numbers This is very important! This law says that So, the more samples we get, the closer the mean is to what is “should” be.

4. Rules for Means If X and Y are random variables, and a and b are fixed numbers, then We will look at these individually in one minute…

5. 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.

6. The other rule 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.

7. Rules for Means Demonstrated X = units sold for military division 10003000500010,000 Probability0.10.30.40.2 Y = units sold for civilian division 300500750 Probability0.40.50.1 If this company makes $2000 for each military unit sold and $3500 on each civilian unit sold, find the mean TOTAL profit.

8. Rules for Variances 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. If X is a random variable, and a and b are fixed numbers, then …

9. 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…

10. What if they are independent? If x and y are independent, they have no effect on one another, so…. ρ = 0 and therefore…

11. Examples… If X and Y are independent random variables and…

12. Putting it all together! One brand of bathtub comes with a dial to set the water temperature. When the “babysafe” setting is selected and the tub is filled, the temperature X of the water follows a Normal distribution with a mean of 34°C and a standard deviation of 2°C. – Define the random variable Y to be the water temperature in degrees Fahrenheit (F = 9/5C + 32) when the dial is set to “babysafe.” Find the mean and standard deviation of Y.

13. Continued… According to Babies R Us, the temperature of a baby’s bathwater should be between 90°F and 100°F. Find the probability that the water temperature on a randomly selected day when the “babysafe” setting is used meets the Babies R Us recommendation.

14. One more! Mr. Starnes likes sugar in his hot tea. From experience, he needs between 8.5 and 9 grams of sugar in his cup of tea for it to taste right. While making his tea one morning, Mr. Starnes adds four randomly selected packets of sugar. Suppose the amount of sugar in these packets follows a Normal distribution with mean 2.17 grams and standard deviation 0.08 grams. What is the probability that Mr. Starnes’ tea tastes right?

15. Homework p. 378 (39, 44, 52 not c, 60, 65, 66)