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5-2 Probability Distributions This section introduces the important concept of a probability distribution, which gives the probability for each value of a variable that is determined by chance. Give consideration to distinguishing between outcomes that are likely to occur by chance and outcomes that are “unusual” in the sense they are not likely to occur by chance. Random Variable Probability Distribution Random Variable a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure Probability Distribution a description that gives the probability for each value of the random variable, often expressed in the format of a graph, table, or formula Discrete and Continuous Random Variables Discrete Random Variable either a finite number of values or countable number of values, where “countable” refers to the fact that there might be infinitely many values, but that they result from a counting process Continuous Random Variable has infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions. Probability Distribution: Requirements 1. There is a numerical random variable x and its values are associated with corresponding probabilities. 2. The sum of all probabilities must be 1. P x 1 3. Each probability value must be between 0 and 1 inclusive. 0 P x 1 Mean, Variance and Standard Deviation of a Probability Distribution [ x P( x)] Mean 2 [( x )2 P( x)] Variance [( x P( x)] Variance 2 2 2 [( x P( x)] 2 2 Standard Deviation Expected Value The expected value of a discrete random variable is denoted by E, and it represents the mean value of the outcomes. It is obtained by finding the value of [ x P( x)]. E [ x P( x)] Example The following table describes the probability distribution for the number of girls in two births. Find the mean, variance, and standard deviation. x P(x) x P x 0 0.25 0.00 0.25 1 0.50 0.50 0.00 2 0.25 0.50 0.25 1.00 0.50 Total x 2 P x Example The following table describes the probability distribution for the number of girls in two births. Find the mean, variance, and standard deviation. Mean x P x 1.0 2 Variance x P x 0.5 2 Standard Deviation 0.5 0.707 Identifying Unusual Results Range Rule of Thumb According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits: Maximum usual value = 2 Minimum usual value = 2 Example – continued We found for families with two children, the mean number of girls is 1.0 and the standard deviation is 0.7 girls. Use those values to find the maximum and minimum usual values for the number of girls. maximum usual value 2 1.0 2 0.7 2.4 minimum usual value 2 1.0 2 0.7 0.4 Identifying Unusual Results Probabilities Rare Event Rule for Inferential Statistics If, under a given assumption (such as the assumption that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is probably not correct. Identifying Unusual Results Probabilities Using Probabilities to Determine When Results Are Unusual Unusually high: x successes among n trials is an unusually high number of successes if P( x or more) 0.05. Unusually low: x successes among n trials is an unusually low number of successes if P( x or fewer) 0.05 .