One place the calculation of µ is useful is in considering the valueof insurance policies.Example: The actuary for an insurance company determines thatyou have a 5% chance in any one year of requiring repair damageto your car, which we will simplify by saying the payout is exactly$2,000. Alternatively, with a 0.1% chance, the car will have to bereplaced completely, and the payout is $20,000. For this policy,the insurance company charges a premium of $150. Is this fair?1In this case let X be “the payout on the policy in a given year”.The possible values and their probabilities are as follows:x P (x) xP (x)0 0.949 02000 0.05 10020000 0.001 20Total 1.00 120So the expected payout per policy is $120. When allowing forthe insurance company’s operating costs and a built-in factor forrisk, the premium of $150 does not seem unreasonable.Real insurance calculations are done a lot like that, but of coursethey are based on many more possible payout scenarios and also,they take into account a lot more individual information (yourage, where you live, record of past claims, etc.)2Spread of a random variableSo far we have only talked about the mean of a random variable,which represents the most natural measure of the center of adistribution. However there are other concepts such as median,mode,...., reflecting our earlier discussion about summarizing thecenter of a sample of data.The spread of a distribution may be measured in numerous ways,of which the standard deviation is the most common. Usuallythe standard deviation is written σ (Greek sigma).We don’t discuss here the formula for computing σ. However, itwill come up when we deal with specific examples later.3Probability Distribution of a Categorical VariableIn principle, categorical variables lie outside the framework ofthis discussion since the outcomes are not numerical, but we canmake them fit by adopting certain conventions. In particular, fora categorical variable with only two outcomes, we can label them0 and 1 to turn them into a discrete numerical variable.Example. In an experiment in which rats are exposed to a toxicsubstance, 35% of them die. We can label the outcomes as 0for alive and 1 for dead. The probability distribution is then:x P (x) xP (x)0 0.65 01 0.35 0.350.35The expected value µ is equal to the probability of the outcomelabelled 1.4Continuous DistributionsA random variable is continuous if it takes a continuum of pos-sible values. For example if X is “height of a person in inches”,there’s no requirement that is be a whole number of inches.• A question like “what is the probability that X = 64?” ismeaningless because nobody has a height exactly 64 inches.• However interval probabilities, such as “what is the probabil-ity that X is between 64 and 65?”, make perfect sense.We often use histograms to summarize the distribution. As theinterval width of the histogram gets smaller and smaller (fora very large sample size), the histogram becomes closer andcloser to a continuous curve. We often think of this curve assummarizing the information in a distribution. The technicalname for this is probability density function.5The Normal DistributionThe normal distribution is said to apply when the shape of thedistribution follows a bell-shaped curve that is unimodal and sym-metric. It is characterized by the mean µ and the standard devi-ation σ.Example: In a sample of men and women it is found that thewomen have a mean height µ = 65 inches and a standard deva-tion σ = 3.5 inches while the men have a mean height µ = 70inches and a standard devation σ = 4.0 inches.In what interval do nearly all of the men lie?Answer: according to the empirical rule this is µ − 3σ to µ + 3σ,or in other words 58 inches to 82 inches.6A more formal statement of this is the 68–95–99.7 rule: If thedata follow a normal distribution, then• 68% of the data lie within one standard deviation of themean• 95% of the data lie within two standard deviations of themean• 99.7% of the data lie within three standard deviations of themean7Example of SAT scores.These are designed to have an overall mean of 500 and a stan-dard deviation of 100.So...• 68% of all students score between 400 and 600• 95% of all students score between 300 and 700• 16% of all students score more than 6008Finding normal probabilitiesA survey shows that women have a mean height of µ = 65 inchesand a standard deviation of σ = 3.5 inches. For a randomlychosen woman, what is the probability that her height is lessthan 70 inches?Step 1: Translate the height x = 70 into an equivalent z-scoreby the formulaz =x − µσ=70 − 653.5= 1.43.Step 2: Look up the answer in the normal probability table (TableA of appendix).In this case, z = 1.43 corresponds to a probability .9236.9Standard Normal Cumulative Probabilitiesz .00 .01 .02 .03 .04 .05 .06 .07 .08 .09-3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002-3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003-3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005-3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007-3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010-2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014-2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019-2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026-2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .0036-2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048-2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064-2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .0084-2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110-2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143-2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183-1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233-1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294-1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367-1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455-1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559-1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694 .0681-1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823-1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .0985-1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170-1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379-0.9 .1841 .1814