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chapter8 - Chapter 8 Discrete Probability Distributions We...

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Chapter 8 Discrete Probability Distributions We begin by defining a random variable . A random variable is a function or rule that assigns a number to each outcome of an experiment. Instead of talking about the coin flipping event as heads, tails, for example. we can think of it as 1 and 0, what we have called an indicator variable. This is actually a numerical event which can be called “the number of heads when flipping a coin.” There are two types of random variables; discrete random variable which take on a countable number of values, for example, sum of the roll of two dice 2,3,...,12), or a continuous random variable that take on values that are uncountable, that is on a real line, for example, time (we can speak of 30.1 minutes, 30.10001 minutes. etc.) A useful analogy is “integers are discrete, while real numbers are continuous.” Probability distributions are a listing of all possible outcomes of an experiment and the corresponding probability. A discrete distribution is based on random variables which can assume only clearly separated values. A continuous distribution can assume an infinite number of values within a given range. All this is not new - we have seen the histogram! 1
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2 CHAPTER 8. DISCRETE PROBABILITY DISTRIBUTIONS 8.1 The discrete random variable When dealing with univariate data, we have seen how the histogram to- gether with its mean and standard deviation are a useful way to summarize data. In statistics, what in fact we are dealing with is a random variable , sometimes denoted r.v. A random variable is a function or rule that assigns a number to each outcome of an experiment. Random variables can be either discrete , i.e one that takes on only whole numbers like 0, 1, 2, ..., or continuous , i.e one that takes on any value on a Real line. A useful analogy is “Integers are discrete, while Real Numbers are continuous.” The outcomes of a random variable can be described by a probability distributions, which are a listing of all possible outcomes of an experiment and their corresponding probabilities. A discrete distribution is based on random variables which can assume only clearly separated values, while a continuous distribution is one that can assume an infinite number of values within a given range. Let us look at an example. Consider an experiment in which a coin is tossed three times. Let X be a r.v. representing the number of heads in 3 tosses. The possible outcomes are: Table 8.1: Outcomes of 3 tosses of a coin Outcome No. of heads T T T 0 T T H 1 T H T 1 T H H 2 H T T 1 H H T 2 H T H 2 H H H 3
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8.1. THE DISCRETE RANDOM VARIABLE 3 Note we have 8 equally likely outcomes from which we can easily con- struct the probability distribution of X , the number of heads in 3 tosses, as: Table 8.2: Probability distribution of 3 tosses of a coin X P(X) 0 1/8 or .125 1 3/8 or .375 2 3/8 or .375 3 1/8 or .125 The probability distribution, sometimes also referred to as the probability function, is in fact a histogram which looks as follows: Figure 8.1: Probability distribution 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 1 2 3 X f(X) The corresponding cumulative probability distribution can be drawn as:
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