Section 6. 1 Probability Rules Objectives- Distinguish between discrete and continuous random variables Identify discrete probability distributions Construct probability histograms Compute and interpret the mean of a discrete random variable Interpret the mean of a discrete random variable as an expected value. Compute the variance and standard deviation of a discrete random variable. Random Variable: A numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are denoted using letters such as X. Discrete Random Variable: Has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point. Continuous Random Variable: Has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion. Probability Distribution: The probability distribution of a discrete random variable X provides the possible values of the random variable and their corresponding probabilities. A probability distribution can be in the form of a table, graph, or mathematical formula. Rules for Discrete Probability Distribution Let P(x) denote the probability that the random variable X equals x; then Sigma P(x)=1 0=P(X)=1 Probability Histogram: A histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of each value of the random variable. The Mean of a Discrete Random Variable The mean of a discrete random variable is given by the formula Mean (x)=Sigma[x*P(x)] where x is the value of the random variable and P(x) is the probability of observing the value x. Interpretation of the Mean of Discrete Random Variable Suppose that an experiment is repeated n independent times and the value of the random variable X is recorded. As the number of repetitions of the experiment increases, the mean value of the n trials will approach Mean(x). IN other words, let x1 be the value of the random variable X after the first experiment, x2 be the random variable X after the second experiment, and so on. Then x bar= x1 + x2 + ? + xn --------------------- n The difference between x bar and Mean(x) gets closer to 0 as n increases. Variance and Standard Deviation of a Discrete Random Variable The variance of a discrete random variable is given by ?2x=Sigma[(x-Mean(x))^2*P(x)] =Sigma [x^2*P(x)]-Mean(x)^2
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