Math 141 WIR, Spring 2007,cBenjamin AurispaMath 141 Key Topics: 8.1-8.3Section 8.1• A random variable is a rule that assigns a number to each outcome of an experiment.• A random variable is called finite discrete if it takes on only finitely many values.• A random variable is called infinite discrete it it takes on infinitely many values which canbe arranged in some order or sequence.• A random variable is called continuous if the values that it can take on make up an intervalof numbers. (Examples are time, distance, or measurements.)• A histogram is a visual representation of the probability distribution of a random variable.The sum of the areas (and heights) of all the rectangles in a histogram equals 1.Section 8.2• The mean or average of a set of numbers x1, x2, . . . , xn, denoted x isx =x1+ x2+ · · · + xnn• The median of a set of numbers is the number that is in the middle when they are arranged inincreasing order if there are an odd number of entries. If there are an even number of entries,the median is the average of the two middle values.• The mode of a set of numbers is the number that occurs the most frequently.• If X is a random variable that takes on the values x1, x2, . . . , xnwith corresponding probabilitiesp1, p2, . . . , pn, then the expected value of X, denoted E(X) isE(X) = x1p1+ x2p2+ · + xnpnVisually, the expected value of a random variable is the point along the x-axis where thehistogram of X would be balanced.• A game is said to be fair if the expected value of the net winnings is 0.• Converting probability to odds: If P (E) is the probability of an event occurring, then– The odds in favor of E occurring areP (E)P (Ec).– The odds against E occurring areP (Ec)P (E).Note: If the odds of an event areab, we say the odds are a to b.• Converting odds to probability: If the odds in favor of E occurring are a to b, then theprobability of E occurring isP (E) =aa + b1Math 141 WIR, Spring 2007,cBenjamin AurispaSection 8.3• The variance of a random variable, V ar(X), is a measure of the spread of the distributionabout the expected value (or mean).• The standard deviation of a random variable is another measure of spread. The standarddeviation, σ, is defined to beσ =qV ar(X) which means that σ2= V ar(X)Note: Not all instructors may cover the following topic.• If X is a random variable with expected value µ and standard deviation σ, then Chebychev’sInequality says thatP (µ − kσ ≤ X ≤ µ + kσ) ≥ 1 −1k2What does this mean? Chebychev’s Inequality allows us to estimate the probability that anoutcome lies within k standard deviations of the mean (expected value). k standard deviationsbelow the mean is µ − kσ and k standard deviations above the mean is µ + kσ. The value1 −1k2is the estimate
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