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Random Variables

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Definitions A random variable is a variable whose value is a numerical outcome of a random phenomenon,. A discrete random variable X has a countable number of possible values. The probability distribution of a discrete random variable X lists the values of their probabilities: ________________________________________________ Value of X:x 1 x 2 x 3...x n Probability:p 1 p 2 p 3...P n __________________________________________________________________________ The following must be true:

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Example 1 NC State posts the grade distributions for its courses online. Students is Statistics 101 in the fall 2003 semester earned 21% A’s, 43% B’s, 30% C’s, and 5% D’s. Choose a students at random. What is the probability that the student failed the course? What is the probability that the student got a B or better?

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Example 2 Spell-checking software catches “non-word errors,” which result in a string of letters that is not a word, as when “the” is typed as “teh.” When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of non-word errors has the following distribution: _______________________________________ X:01234 P(x):0.10.20.30.30.1 ____________________________________________________________ Write the event “at least one non-word error” in terms of X. What is the probability of this event? Describe the event X ≤ 2 in words. What is its probability? What is the probability that X < 2?

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Mean of a Discrete Random Variable Suppose that X is a discrete random variable whose distribution is ________________________________________________ Value of X:x 1 x 2 x 3...x n Probability:p 1 p 2 p 3...P n __________________________________________________________________________ To find the mean of X, multiply each possible value by its probability, then add all the products.

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Example 1 Revisited NC State posts the grade distributions for its courses online. Students is Statistics 101 in the fall 2003 semester earned 21% A’s, 43% B’s, 30% C’s, 5% D’s and 1% F’s. Choose a students at random. What is the mean grade for this course?

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Example 2 Revisited Spell-checking software catches “non-word errors,” which result in a string of letters that is not a word, as when “the” is typed as “teh.” When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of non-word errors has the following distribution: _______________________________________ X:01234 P(x):0.10.20.30.30.1 ____________________________________________________________ What is the expected number of non-word errors in the 250-word essay?

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Example 3 A single male driver aged 23 living in a suburb of a Midwestern city pays a car insurance premium of $496 per year. The insurance company’s actuarial model for this driver predicts the following probabilities for 1 year: 0.065 for an accident averaging $4600 in damages; 0.03 for $2400 in damages; and 0.015 for $1000 in damages. What is the expected damage cost the insurance company should be prepared to pay for such a driver? What amount is the policy expected to contribute to the operation and profit of the company?

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Example 4 Two coins are tossed. If both land heads up, the player A wins $4 from Player B. If exactly one coin lands heads up, then Player b wins $1 from Player A. If both land tails up, then Player B wins $2 from Player A. Is this a fair game?

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Example 5 You pick a number from 1 – 6 and roll 3 dice. If the number you pick comes up 3 times, you win $3. If it comes up twice, you win $2, and once you win $1, otherwise you lose $1. What is your expected gain or loss?

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