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Chapter 6. Random Variables & Distributions

Chapter 6. Random Variables & Distributions - 6 6.1...

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§ 6 Random variables and distributions § 6.1 Random variable 6.1.1 Consider an experiment with sample space Ω. Definition. (Informal) A random variable is a function X : Ω ( -∞ , ). 6.1.2 A random variable is useful when we want to quantify experimental outcomes or descriptive events by real numbers. Sometimes, it may be convenient to simplify the sample space so that only those events of interest are described by X . 6.1.3 Examples. (i) Ω = { win, lose } can be transformed by defining X (win) = 1, X (lose) = 0. (ii) Toss 2 coins. Ω = { HH, HT, TH, TT } . Interested in no. of heads only. Define X (HH) = 2 , X (HT) = X (TH) = 1 , X (TT) = 0 . (iii) Toss 2 coins. Ω = { HH, HT, TH, TT } . Interested in no. of vertical strokes in each pair of letters. Define X (HH) = 4 , X (HT) = X (TH) = 3 , X (TT) = 2 . (iv) n Bernoulli trials. Interested in no. of successes. Define X = no. of successes. (v) Annual income Y has sample space [0 , ). Taxable when annual income exceeds c say. Only interested in part of income which is taxable. May define X = ( 0 , Y c, Y - c, Y > c. 6.1.4 Conventional notation : Use capital letters X, Y, . . . to denote random variables and small letters x, y, . . . the possible numerical values (or realizations ) of these variables. 27
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§ 6.2 Distribution function 6.2.1 Definition. The distribution function of a random variable X is the function F : ( -∞ , ) [0 , 1] given by F ( x ) = P ( X x ) . N.B. Alternative name: cumulative distribution function (cdf). 6.2.2 Example. Toss a coin twice, with Ω = { HH, HT, TH, TT } . Define X = no. of heads; Y = 1 if both tosses return the same side, and = - 1 otherwise. Distribution function of X : F X ( t ) = 0 , t < 0 , 1 / 4 , 0 t < 1 , 3 / 4 , 1 t < 2 , 1 , t 2 . Y : F Y ( t ) = 0 , t < - 1 , 1 / 2 , - 1 t < 1 , 1 , t 1 . 6.2.3 The following properties characterize a cdf: (i) lim x →-∞ F ( x ) = 0 (ii) lim x →∞ F ( x ) = 1 (iii) F is increasing. (iv) F is right-continuous , i.e. lim h 0 F ( x + h ) = F ( x ). N.B. F is not necessarily left-continuous: see example above. 6.2.4 Random variables and distribution functions are useful in describing probability models. The probabilities attributed to events concerning a random variable X can be reconstructed from the distribution function of X . 28
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Example. If X denotes a Binomial ( n, p ) random variable and its distribution function F is given, then we can calculate P ( X = r ) = F ( r ) - F ( r - 1) , r = 0 , 1 , 2 , . . . , n, and deduce from these the probability of any event involving X . § 6.3 Discrete random variables 6.3.1 Definition. Let X be a random variable defined on the sample space Ω. Then X is a discrete random variable if X (Ω) ≡ { X ( ω ) : ω Ω } is countable . N.B. A set A is countable if its elements can be enumerated (or listed) as { a 1 , a 2 , . . . } . 6.3.2 Examples. (i) Binomial ( n, p ): X (Ω) = { 0 , 1 , 2 , . . . , n } . (ii) Bernoulli trial: X (Ω) = { 0 , 1 } .
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