ST 370 Probability and Statistics for Engineers Discrete Random Variables A random variable is a numerical value associated with the outcome of an experiment. Discrete random variable When we can enumerate the possible values of the variable (such as 0, 1, 2, . . . ), the random variable is discrete . Example: acceptance sampling Suppose that a sample of size 10 is drawn from a shipment of 200 items, of which some number are non-compliant; X is the number of non-compliant items in the sample. The possible values of X are 0, 1, 2, . . . , 10, so X is a discrete random variable. 1 / 15 Discrete Random Variables
ST 370 Probability and Statistics for Engineers Continuous random variable When the variable takes values in an entire interval, the random variable is continuous . Example: flash unit recharge time Suppose that a cell phone camera flash is chosen randomly from a production line; the time X that it takes to recharge is a positive real number; X is a continuous random variable. Presumably, there is some lower bound a > 0 that is the shortest possible recharge time, and similarly some upper bound b < ∞ that is the longest possible recharge time; however, we usually do not know these values, and we would just say that the possible values of X are { x : 0 < x < ∞} . 2 / 15 Discrete Random Variables
ST 370 Probability and Statistics for Engineers Probability distribution The probability distribution of a random variable X is a description of the probabilities associated with the possible values of X . The representation of a probability distribution is different for discrete and continuous random variables. Probability mass function For a discrete random variable, the simplest representation is the probability mass function (pmf) f X ( x ) = P ( X = x ) where x is any possible value of X . 3 / 15 Discrete Random Variables Probability distribution
ST 370 Probability and Statistics for Engineers Example: acceptance sampling Suppose one item is chosen at random from a shipment of 200 items, of which 5 are non-compliant. Let � 1 if the item is non-compliant, X = 0 if the item is compliant. We could say that X is the number of non-compliant items seen. The probability mass function of X is � 0 . 975 x = 0 f X ( x ) = 0 . 025 x = 1 A random variable like X that takes only the values 0 and 1 is called a Bernoulli random variable. 4 / 15 Discrete Random Variables Probability distribution
ST 370 Probability and Statistics for Engineers Example: Dice Suppose you roll a fair die, and the number of spots showing is X . Then X is a discrete random variable with probability mass function f X ( x ) = 1 6 , x = 1 , 2 , 3 , 4 , 5 , 6 . Because the probability is the same for all the possible values of X , it is called the discrete uniform distribution. Properties of the probability mass function They are probabilities: f X ( x ) ≥ 0. They cover all possibilities: � x f X ( x ) = 1. 5 / 15 Discrete Random Variables Probability distribution
ST 370 Probability and Statistics for Engineers Cumulative distribution function As an alternative to the probability mass function, the probability distribution of a random variable X can be defined by its cumulative distribution function (cdf) F X ( x ) = P ( X ≤ x ) , −∞ < x < ∞ . In terms of the probability mass function: � F X ( x ) = f x ( x i ) , −∞ < x < ∞ . x i ≤ x F X ( · ) has a jump at each possible value x i of X , and the jump equals the corresponding probability f X ( x i ), so the probability mass function can be obtained from the cumulative distribution function. 6 / 15 Discrete Random Variables Cumulative distribution function
ST 370 Probability and Statistics for Engineers Example: acceptance sampling 0 x < 0 F X ( x ) = 0 . 975 0 ≤ x < 1 1 x ≥ 1 curve(pbinom(x, 1, .025), from = -1, to = 2) 7 / 15 Discrete Random Variables Cumulative distribution function
ST 370 Probability and Statistics for Engineers Example: dice 0 x < 1 1 / 6 1 ≤ x < 2 2 / 6 2 ≤ x < 3 F X ( x ) = . . . 1 x ≥ 6 curve(pmax(0, pmin(1, floor(x)/6)), from = 0, to = 7) 8 / 15 Discrete Random Variables Cumulative distribution function
ST 370 Probability and Statistics for Engineers Mean and Variance Mean value The mean value , or expected value, of a discrete random variable with probability mass function f X ( · ) is � µ X = E ( X ) = xf X ( x ) . x E ( X ) is a weighted average of the possible values of X , each weighted by the corresponding probability. The expected value E ( X ) is a typical value of the random variable X , in the same way that a sample mean ¯ x is a typical value of the sample x 1 , x 2 , . . . , x n . 9 / 15 Discrete Random Variables Mean and Variance
ST 370 Probability and Statistics for Engineers Example: acceptance sampling One item is chosen at random from a shipment of 200 items, of which 5 are non-compliant, and X is the number of non-compliant items seen: E ( X ) = 0 × f X (0) + 1 × f X (1) = 0 . 025 . For any Bernoulli random variable X , E ( X ) = P ( X = 1) . 10 / 15 Discrete Random Variables Mean and Variance
ST 370 Probability and Statistics for Engineers Example: Dice Suppose you roll a fair die, and the number of spots showing is X : E ( X ) = 1 × f X (1) + 2 × f X (2) + · · · + 6 × f X (6) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3 . 5 . Note In these examples and in many others, the “expected” value is not one of the possible values of the random variable; this is not the paradox that it is sometimes made out to be! 11 / 15 Discrete Random Variables Mean and Variance
ST 370 Probability and Statistics for Engineers Variance Suppose that X is a random variable with expected value µ X . Then Y = ( X − µ X ) 2 is another random variable, and its expected value is � E ( Y ) = yf Y ( y ) y � ( x − µ X ) 2 f X ( x ) . = x 12 / 15 Discrete Random Variables Mean and Variance
ST 370 Probability and Statistics for Engineers The variance of X is E ( Y ) = E [( X − µ X ) 2 ]: σ 2 ( X − µ X ) 2 � � X = V ( X ) = E . The standard deviation of X is � σ 2 σ X = X . 13 / 15 Discrete Random Variables Mean and Variance
ST 370 Probability and Statistics for Engineers Example: acceptance sampling For any Bernoulli random variable X , µ X = P ( X = 1) = p , say , so X = (0 − p ) 2 × P ( X = 0) + (1 − p ) 2 × P ( X = 1) σ 2 = p 2 (1 − p ) + (1 − p ) 2 p = p (1 − p ) and � σ X = p (1 − p ) . 14 / 15 Discrete Random Variables Mean and Variance
ST 370 Probability and Statistics for Engineers Example: Dice ( x − 3 . 5) 2 ( x − 3 . 5) 2 f X ( x ) x x − 3 . 5 f X ( x ) 1 1 -2.5 6.25 1.0417 6 1 2 -1.5 2.25 0.3750 6 1 3 -0.5 0.25 0.0417 6 1 4 0.5 0.25 0.0417 6 1 5 1.5 2.25 0.3750 6 1 6 2.5 6.25 1.0417 6 Total: 2.9168 So σ 2 X = 2 . 917 and σ X = 1 . 708. 15 / 15 Discrete Random Variables Mean and Variance
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