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Discrete Random Variables and Their Probability Distributions Dr. Tai-kuang Ho, National Tsing Hua University The slides draw from the textbooks of Wackerly, Mendenhall, and Schea¤er (2008) & Devore and Berk (2012) 1 3.1 Basic De…nition Numerical events Let Y denote a variable to be measured in an experiment. To each sample point in the sample space S we will assign a real number denoting the value of the variable Y . The sample space S can be partitioned into subsets so that points within a subset are all assigned the same value of Y . 2 These subsets are mutually exclusive since no point is assigned two di¤erent numerical values. We convert an event to a number for simpli…cation and convenience. Figure 2.14 Random variable A random variable is a real-valued function de…ned over a sample space. 3 A random variable is a rule of association between experiment outcomes and numerical values. Figure 3.1 (B.S.) A random variable Y is said to be discrete if it can assume only a …nite or countably in…nite number of distinct values. Examples? Number of voters favoring a certain candidate 4 Number of defect TV sets Number of bacteria per unit area 3.2 The Probability Distribution for a Discrete Random Variable Y : denote a random variable y : denote a particular value Y is a random variable, but the speci…c observed value y is not random. 5 P (Y = y ) De…nition: probability function The probability that Y takes on the value y , P (Y = y ), is de…ned as the sum of the probabilities of all sample points in S that are assigned the value y . We will sometimes denote P (Y = y ) by p (y ). De…nition: probability distribution The probability distribution for a discrete variable Y can be represented by a formula, a table, or a graph that provides p (y ) = P (Y = y ) for all y . De…nition: probability distribution B.S. The probability mass function (pmf) of a discrete random variable is de…ned for every number x by p (x) = P (X = x) = P (all s 2 S : X (s) = x). 6 Example 3.1 Three methods to present a probability distribution: table, histogram, and formula. p (y ) = 3 y ! 3 2 6 2 ! Table 3.1 7 y ! ; y = 0; 1; 2 Figure 3.1 Discrete probability distribution must satis…ed the properties of probability. Theorem: properties of discrete probability distribution For any discrete probability distribution, the following must be true: 1. 0 p (y ) 1 for all y . P 2. y p (y ) = 1, where the summation is over all values of y with nonzero probability. More examples: Figure 3.2 (B.S.); Figure 3.3 (B.S.) 8 The Cumulative Distribution Function (cdf) De…nition: cumulative distribution function The cumulative distribution function F (x) of a discrete random variable X with pmf p (x) is de…ned for every number x by F (x) = P (X x) = y; X p (y ) y x For any number x, F (x) is the probability that the observed value of X will be at most x. 9 Example 3.11 (B.S.); Figure 3.5 (B.S.) Properties of cdf. Cumulative Distribution Function Let X be a random variable. Then its cumulative distribution function (cdf) is de…ned by: FX (x) = PX ([ 1; x]) = P (X 10 x) Let X be a random variable with cdf F (x). Then 1. For all a and b, if a < b then F (a) F (b). 2. limx! 1 F (x) = 0 3. limx!1 F (x) = 1 4. limx#x0 F (x) = F (x0), F is right continuous For any random variable, 11 P [X = x] = FX (x) FX (x ) ; for all x 2 R, where FX (x ) = limz"x FX (z ). 3.3 The Expected Value of a Random Variable or a Function of a Random Variable De…nition: expected value Let Y be a discrete random variable with the probability function p (y ). Then the expected value of Y , E (Y ), is de…ned to be 12 E (Y ) = X y p (y ) y E (Y ) = Table 3.2 Figure 3.2 1 1 1 +1 +2 =1 E (Y ) = 0 4 2 4 13 Theorem: expected value of a function Let Y be a discrete random variable with probability function p (y ) and g (Y ) be a real-valued function of Y . Then the expected value of g (Y ) is given by E [g (Y )] = X g (y ) p (y ) allY De…nition: variance of a random variable If Y is a random variable with mean E (Y ) = , the variance of a random variable Y is de…ned to be the expected value of (Y )2. That is, 14 h V (Y ) = E (Y ) 2 i V (Y ) = 2 The standard deviation of Y is the positive square root of V (Y ). Example 3.2 Table 3.3 15 Figure 3.3 Some additional tools Theorem: expectation of a constant Let Y be a discrete random variable with probability function p (y ) and c be a constant. Then E (c) = c. Theorem Let Y be a discrete random variable with probability function p (y ), g (Y ) be a function of Y , and c be a constant. Then E [c g (Y )] = c E [g (Y )] 16 Theorem Let Y be a discrete random variable with probability function p (y ) and g1 (Y ), g2 (Y ), . . . , gk (Y ) be k functions of Y . Then E [g1 (Y ) + g2 (Y ) + + gk (Y )] = E [g1 (Y )] + E [g2 (Y )] + Theorem E (aY + b) = a E (Y ) + b Theorem E (aY ) = a E (Y ) Theorem E (Y + b) = E (Y ) + b 17 + E [gk (Y )] Theorem: variance and second moment Let Y be a discrete variable with probability function p (y ) and mean E (Y ) = ; then V (Y ) = 2 h = E (Y ) 2 i =E Y2 2 Proof 2 h 2 i =E Y2 2 Y + 2 = E (Y ) = E Y2 2 E (Y ) + 2 = E Y 2 18 2 2+ 2 Theorem V ar (aY + b) = a2 V ar (Y ) Theorem V ar (Y + b) = V ar (Y ) 3.4 The Binomial Probability Distribution A Bernoulli experiment is a random experiment, the outcome of which can be classi…ed in but one of two mutually exclusive and exhaustive ways. For example, success or failure, female or male, life of death, non-defective or defective. 19 Let X be a random variable associated with a Bernoulli experiment. X (success) = 1 X (f ailure) = 0 What is the parameter of a Bernoulli distribution? p De…nition: parameter of a distribution Suppose p depends on a quantity that can be assigned any one of a number of possible values, with each di¤erent value determining a di¤erent probability distribution. Such a quantity is called a parameter of the distribution. 20 The probability mass function (pmf) of X can be written as: p (x) = px (1 p)1 x ; x = 0; 1 The expected value of X and the variance of X is: = E (X ) = 0 (1 2 h = E (X ) 2 i = p2 (1 p) + 1 p = p p) + (1 21 p)2 p = p (1 p) A sequence of Bernoulli experiments occur when a Bernoulli experiment is performed several independent times so that the probability of success, say p, remains the same from experiment to experiment. toss a coin n times shot in a sequence of …rings at a target De…nition: Binomial Experiment A binomial experiment possesses the following properties: 22 1. The experiment consists of a …xed number n of identical trials. 2. Each trial results in one of two outcomes: success S or failure F . 3. The probability of success on a single trial is equal to some value p and remains the same from trial to trial. The probability of a failure is equal to q = (1 p). 4. The trials are independent. 5. The random variable of interest is Y , the number of successes observed during the n trials. 23 The random variable of interest in a Bernoulli experiment is success or failure. Important: the random variable of interest in the binomial experiment is the number of successes observed in the n experiments (trials). A random sample of n = 10 voters will be selected, and Y , the number of favoring some candidate is to be observed. This experiment approximates a binomial experiment. Binomial Probability Distribution De…nition: pmf of a binomial distribution A random variable Y is said to have a binomial distribution based on n trials with success probability p if and only if 24 p (y ) = n y ! n 1 ! py q n y ; y = 0; 1; 2; : : : ; n Figure 3.4 (p + q )n = | n 0 ! {z p(0) qn + } | p1 q n 1 + {z p(1) } 25 n 2 ! p2 q n 2 + + | n n ! {z p(n) pn } X y p (y ) = n X y=0 n y ! py q n y = (p + q )n = 1 Draw a picture of binomial distribution using R. vals=dbinom(1:10,size=10,prob=0.2) barplot(vals,names=1:10,main="Binomial(n=10,p=0.2)") What are the parameters of the distribution? 26 p; n Mean and variance of binomial distribution Theorem: mean and variance of binomial distribution Let Y be a binomial random variable based on n trials and success probability p. Then = E (Y ) = np 2 = V (Y ) = npq 27 Please work out the proof at home. Most texts use moment generating function to compute mean and variance of binomial distribution. 28