IPS6eCh04_3_4bb

# IPS6eCh04_3_4bb - Probability The Study of Randomness...

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Probability: The Study of  Randomness Random Variables IPS Chapters 4.3 and 4.4 © 2009 W.H. Freeman and Company

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Objectives (IPS Chapters 4.3 and 4.4) Random variables Discrete random variables Continuous random variables Normal probability distributions Mean of a random variable Law of large numbers Variance of a random variable Rules for means and variances
Discrete random variables A random variable is a variable whose value is a numerical outcome of a random phenomenon. A basketball player shoots three free throws. We define the random variable X as the number of baskets successfully made. A discrete random variable X has a finite number of possible values. A basketball player shoots three free throws. The number of baskets successfully made is a discrete random variable ( X ). X can only take the values 0, 1, 2, or 3.

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The probability distribution of a random variable X lists the values and their probabilities: The probabilities p i must add up to 1. A basketball player shoots three free throws. The random variable X is the number of baskets successfully made. Value of X 0 1 2 3 Probability 1/8 3/8 3/8 1/8 H - H HMM HHM MHM HMH MMM MMH MHH HHH
The probability of any event is the sum of the probabilities p i of the values of X that make up the event. A basketball player shoots three free throws. The random variable X is the number of baskets successfully made. Value of X 0 1 2 3 Probability 1/8 3/8 3/8 1/8 HMM HHM MHM HMH MMM MMH MHH HHH What is the probability that the player successfully makes at least two baskets (“at least two” means “two or more”)? P ( X ≥2) = P ( X =2) + P ( X =3) = 3/8 + 1/8 = 1/2 What is the probability that the player successfully makes fewer than three baskets? P ( X <3) = P ( X =0) + P ( X =1) + P ( X =2) = 1/8 + 3/8 + 3/8 = 7/8 or P ( X <3) = 1 – P ( X =3) = 1 – 1/8 = 7/8

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A continuous random variable X takes all values in an interval . Example: There is an infinity of numbers between 0 and 1 (e.g., 0.001, 0.4, 0.0063876). How do we assign probabilities to events in an infinite sample space? We use density curves and compute probabilities for intervals . The probability of any event is the area under the density curve for the values of X that make up the event. Continuous random variables
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IPS6eCh04_3_4bb - Probability The Study of Randomness...

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