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Chapter 16 Random Variables

Intro example. Ex. An insurance company offers a disability policy that pays $10,000 if you lose your eyes or ears or $5000 if you lose your nose :P. How can we determine how much, on average, the company should expect to pay out to a policyholder?. Terminology. We need to build a probability model

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Chapter 16 Random Variables

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    1. Chapter 16 – Random Variables Definition & Types Probability Model / Distribution Mean (expected Value) Standard Deviation

    2. Intro example Ex. An insurance company offers a disability policy that pays $10,000 if you lose your eyes or ears or $5000 if you lose your nose :P. How can we determine how much, on average, the company should expect to pay out to a policyholder?

    3. Terminology We need to build a probability model to answer the previous question The amount the company pays out on an individual policy is called a random variable because the value is based on the outcome of a random event Use a capital letter to denote a random variable… then we use the lowercase version of that same letter to represent a particular value For example, if X = the payout on an insurance policy, then an individual policy could have x = $10,000 or x = $5,000

    4. Discrete vs Continuous If we can list all the outcomes of a random variable, we say that random variable is discrete If the random variable can take on an infinite number of values, we describe it as continuous

    5. Probability Model The collection of all possible values for a random variability and the corresponding probabilities is called the probability model for the random variable Suppose the probability that someone loses eyes or ears is 1 / 1000, and that the probability of losing a nose is 1 / 500 (or 2 / 1000)

    6. Our model Let X = the payout for a policyholder x $10,000 $5,000 $0 P(X = x) 1/1000 2/1000 997/1000 So if the company has 1000 customers, we expect to make one $10,000 payout and two $5,000 payments, so that the average payout for each policy is: 20,000 / 1000 = $20 payout per customer This means that the company will probably need to charge over $20 per policy to stay in business

    7. Expected value We can’t predict with certainty what will happen in a given year, but we can say what we expect to happen The expected value of a policy is a parameter of our probability model, and it is equivalent to the mean of the probability model We can use µ for the mean of the model or we can use E(X) for the expected value, but they are both calculated using the same formula: µ = E(X) = x * P(X = x)

    8. So for our example… E(X) = 10,000 * (1/1000) + 5000 * (2/1000) + 0 * (997/1000) = $20

    9. What about the spread of our model? The other parameter we use to describe a probability model is the standard deviation Recall that the standard deviation is the square root of a quantity called…? The variance! For a probability model, the variance is given by: s2 = Var(X) = To find the standard deviation, take the square root!! s =

    10. In our example… s2 = (10,000 – 20)2 *1/1000 + (5,000 – 20) 2 *2/1000 + (0 – 20)2 *997/1000 = 149,600 square dollars So s is the square root of 149,600 which is approximately $386.78 Note that the standard deviation is HUGE compared to the mean of $20.. This is because of the few HUGE payouts that are so far above the mean payout!!!

    11. HOMEWORK!!!! :P ? ? ? Pg. 321 #s 1, 3, 9, 19, 21

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