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Topic 7 Random Variables and Distribution Functions Distribution Functions Definition of a Random Variable Distribution Functions Properties of Distribution Functions Topic 7 Random Variables and Distribution Functions Distribution Functions 1 / 11 Definition of a Random Variable Distribution Functions Properties of Distribution Functions Outline Definition of a Random Variable Distribution Functions Discrete Random Variables Continuous Random Variables Properties of Distribution Functions 2 / 11 Definition of a Random Variable Distribution Functions Properties of Distribution Functions Introduction statistics probability universe of sample space -Ω information and probability - P ask a question+ and define a+ random collect data variable X organize+ into the organize+ into the empirical cumulative cumulative distribution function distribution function compute+ sample compute distributional+ means and variances means and variances 3 / 11 Definition of a Random Variable Distribution Functions Properties of Distribution Functions Definition of a Random Variable A random variable is a real valued function from the probability space. X :Ω R: ! Typically, we shall use capital letters near the end of the alphabet, e.g., X ; Y ; Z for random variables. The range of a random variable is called the state space. Exercise. Give some random variables on the following probability spaces,Ω. 1. Roll a die3 times and consider the sample space Ω = (i; j; k); i; j; k = 1; 2; 3; 4; 5; 6 . 2. Flip af coin 10 times and consider theg sample spaceΩ, the set of 10-tuples of heads and tails. We can create new random variables via composition of functions: ! X (!) g(X (!)) 7! 7! Thus, if X is a random variable, then so are p X 2; exp αX ; X 2 + 1; tan2 X ; X b c 4 / 11 Definition of a Random Variable Distribution Functions Properties of Distribution Functions Distribution Functions A (cumulative) distribution function of a random variable X is defined by FX (x) = P ! Ω; X (!) x = P X x : f 2 ≤ g f ≤ g For the complement of X x , we have the survival function f ≤ g F X (x) = P X > x = 1 P X x = 1 FX (x): f g − f ≤ g − Choose a < b, then the event X a X b . Their set theoretic difference f ≤ g ⊂ f ≤ g X b X a = a < X b : f ≤ g n f ≤ g f ≤ g Consequently, by the difference rule for probabilities, P a < X b = P( X b X a ) = P X b P X a = FX (b) FX (a): f ≤ g f ≤ g n f ≤ g f ≤ g − f ≤ g − In particular, FX is non-decreasing. 5 / 11 Introduction to the Science of Statistics Random Variables and Distribution Functions Exercise 7.7. For a random variable X and subset B of the sample space S, define P (B)=P X B . X { 2 } Show that PX is a probability. For the complement of X x , we have the survival function { } F¯ (x)=P X>x =1 P X x =1 F (x). X { } − { } − X Choose a<b, then the event X a X b . Their set theoretic difference { } ⇢ { } X b X a = a<X b . { }\{ } { } In words, the event that X is less than or equal to b but not less than or equal to a is the event that X is greater than a and less than or equal to b. Consequently, by the difference rule for probabilities, Definition of a Random Variable Distribution Functions Properties of Distribution Functions P a<X b = P ( X b X a )=P X b P X a = F (b) F (a). (7.2) { } { }\{ } { } − { } X − X Thus, we can compute the probability that a random variable takes values in an interval by subtracting the distri- bution function evaluated at the endpointsDistribution of the intervals. Care Functions is needed on the issue of the inclusion or exclusion of the endpoints of the interval. Let X be the sum of the values on two fair dice, Example 7.8. To give the cumulative distribution function for X, the sum of the values for two rolls of a die, we start x with the2 table 3 4 5 6 7 8 9 10 11 12 x 2 3 4 5 6 7 8 9 10 11 12 P X = x 1/36P X 2/36= x 1/36 3/36 2/36 3/36 4/36 4/36 5/36 5/36 6/36 6/36 5/36 4/36 5/36 3/36 2/364/36 1/36 3/36 2/36 1/36 f g { } Here is FX andthe create cumulative the graph. distribution function. 1 6 r r r 3/4 r r 1/2 r r 1/4 r r - r 1 2r 3 4 5 6 7 8 9 10 11 12 Figure 7.1: Graph of FX , the cumulative distribution function for the sum of the values for two rolls of a die. 6 / 11 103 Definition of a Random Variable Distribution Functions Properties of Distribution Functions Distribution Functions Notice that the distribution function is constant in between the possible values for X , • has a jump size at x is equal to P X = x , and • f g is right continuous. • Call X a discrete random variable if its distribution function FX has these properties. Examples. 3 6 3 = P X = 4 = FX (4) FX (4 ) = 36 f g − − 36 − 36 21 6 15 5 P 4 < X 7 = FX (7) FX (4) = = = : f ≤ g − 36 − 36 36 12 21 3 18 1 P 4 X 7 = FX (7) FX (4 ) = = = : f ≤ ≤ g − − 36 − 36 36 2 7 / 11 Definition of a Random Variable Distribution Functions Properties of Distribution Functions Distribution Functions Exercise. 1. Flip a fair coins 3 times. Let X be the number of heads. Under equally likely outcomes, find P X = x for x = 0; 1; 2; and3 : f g and use this to sketch a graph of the distribution function FX . 2. Deal 5 cards out of a deck of 52. Let X be the number of . Under equally likely } outcomes, use the choose function in R to determine P X = x for x = 0; 1; 2; 3; 4; and5 : f g and use this to sketch a graph of the distribution function FX . 8 / 11 Definition of a Random Variable Distribution Functions Properties of Distribution Functions Distribution Functions 1 For a dart board with radius 1, assume that 0.8 the dart lands randomly uniformly. Let X be 0.6 0.4 the distance from the center. For x [0; 1], 0.2 0 2 x !0.2 FX (x) = P X x !0.4 f ≤ g !0.6 area inside circle of radius x !0.8 = !1 area of circle 2 !1 !0.8 !0.6 !0.4 !0.2 0 0.2 0.4 0.6 0.8 1 πx 2 = 2 = x : π1 1.0 Thus, we have the distribution function 0.8 0.6 probability 8 0.4 < 0 if x 0; 2 ≤ 0.2 FX (x) = x if0 < x 1; ≤ 0.0 : 1 if x > 1: 0.0 0.5 1.0 x 9 / 11 Definition of a Random Variable Distribution Functions Properties of Distribution Functions Distribution Functions Exercise. 1. Find the probability that the dart no more than 1/2 unit from the center. 2. Find the probability that the dart lands further 1/3 unit but no more than 2/3 unit from the center. 3. Find the median, x so that P X x = 1=2. 1=2 f ≤ 1=2g Definition. X is continuous random variable if it has a cumulative distribution function FX that is differentiable. 10 / 11 Definition of a Random Variable Distribution Functions Properties of Distribution Functions Properties of Distribution Functions A distribution function FX has the property that it is right continuous, starts at0 , ends at1, and does not decrease with increasing values of x. 1.0 In mathematical terms, For every a, lim F (x) = F (a). 0.8 • x!a+ X X lim F (x) = 0. 0.6 • x→−∞ X lim F (x) = 1. x!1 X 0.4 • distribution function For every a; b satisfying a < b, • 0.2 FX (a) Fx (b): ≤ 0.0 0 5 10 15 20 25 x 11 / 11.
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