Chapter 5 Discrete Distributions

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108 CHAPTER 5. DISCRETE DISTRIBUTIONS Every discrete random variable X has associated with it a probability mass function (PMF) f X : S X → [0, 1] defined by f X (x) = IP(X = x), x ∈ S X . (5.1.2) Since values of the PMF represent probabilities, we know from <strong>Chapter</strong>4 that PMFs enjoy certain properties. In particular, all PMFs satisfy 1. f X (x) > 0forx ∈ S , 2. ∑ x∈S f X (x) = 1, and > should be >= and S should be S_X 3. IP(X ∈ A) = ∑ x∈A f X (x), for any event A ⊂ S . Delete text from "for" on Example 5.1. Toss a coin 3 times. The sample space would be S = {HHH, HTH, THH, TTH, HHT, HTT, THT, TTT} . Now let X be the number of Heads observed. Then X has support S X = {0, 1, 2, 3}. Assuming that the coin is fair and was tossed in exactly the same way each time,itisnotunreasonable to suppose that the outcomes in the sample space are all equally likely. What is the PMF of X NoticethatX is zero exactly when the outcome TTT occurs, and this event has probability 1/8. Therefore, f X (0) = 1/8, and the same reasoning shows that f X (3) = 1/8. Exactly three outcomes result in X = 1, thus, f X (1) = 3/8 and f X (3) holds the remaining 3/8probability(the total is 1). We can represent the PMF with a table: x ∈ S X 0 1 2 3 Total f X (x) = IP(X = x) 1/8 3/8 3/8 1/8 1 5.1.2 Mean, Variance, and Standard Deviation There are numbers associated with PMFs. One important example is the mean µ, alsoknown as IE X: ∑ µ = IE X = xf X (x), (5.1.3) provided the (potentially infinite) series ∑ |x| f X (x)isconvergent.Anotherimportantnumberis the variance: ∑ σ 2 = IE(X − µ) 2 = (x − µ) 2 f X (x), (5.1.4) which can be computed (see Exercise 5.4) withthealternateformulaσ 2 = IE X 2 − (IE X) 2 . Directly defined from the variance is the standard deviation σ = √ σ 2 . Example 5.2. We will calculate the mean of X in Example 5.1. 3.5 should be 1.5 3∑ µ = xf X (x) = 0 · 1 8 + 1 · 3 8 + 2 · 3 8 + 3 · 1 8 = 3.5. x=0 We interpret µ = 3.5byreasoningthatifweweretorepeattherandomexperimentmany times, independently each time, observe many corresponding outcomes of the random variable X,and take the sample mean of the observations, then the calculatedvaluewouldfallcloseto3.5.The approximation would get better as we observe more and more values of X (another form of the Law of Large Numbers; see Section 4.3). Another way it is commonly stated is that X is 3.5 “on the average” or “in the long run”. x∈S x∈S
5.1. DISCRETE RANDOM VARIABLES 109 Remark 5.3. Note that although we say X is 3.5 on the average, we must keep in mind that our X never actually equals 3.5 (in fact, it is impossible for X to equal 3.5). Related to the probability mass function f X (x) = IP(X = x) isanotherimportantfunction called the cumulative distribution function (CDF), F X .Itisdefinedbytheformula F X (t) = IP(X ≤ t), −∞ < t < ∞. (5.1.5) We know that all PMFs satisfy certain properties, and a similar statement may be made for CDFs. In particular, any CDF F X satisfies • F X is nondecreasing (t 1 ≤ t 2 implies F X (t 1 ) ≤ F X (t 2 )). • F X is right-continuous (lim t→a + F X (t) = F X (a)foralla ∈ R). • lim t→−∞ F X (t) = 0andlim t→∞ F X (t) = 1. We say that X has the distribution F X and we write X ∼ F X .Inanabuseofnotationwewill also write X ∼ f X and for the named distributions the PMF or CDF will be identified by the family name instead of the defining formula. Actually, the ~ notation is standard only for distributions specified by name and parameters 5.1.3 How to do it with R The mean and variance of a discrete random variable is easy to compute at the console. Let’s return to Example 5.2. Wewillstartbydefiningavectorx containing the support of X, anda vector f to contain the values of f X at the respective outcomes in x: > x f mu mu [1] 1.5 To compute the variance σ 2 ,wesubtractthevalueofmu from each entry in x, squarethe answers, multiply by f, andsum. Thestandarddeviationσ is simply the square root of σ 2 . > sigma2 sigma2 [1] 0.75 > sigma sigma [1] 0.8660254 Finally, we may find the values of the CDF F X on the support by accumulating the probabilities in f X with the cumsum function.
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