Topic 7 Random Variables and Distribution Functions

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1 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Topic 7 Random Vaiables and Distibution Functions Distibution Functions 1 / 11

2 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Outline Definition of a Random Vaiable Distibution Functions Discete Random Vaiables Continuous Random Vaiables Popeties of Distibution Functions 2 / 11

3 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Intoduction statistics univese of infomation ask a question and collect data oganize into the empiical cumulative distibution function compute sample means and vaiances pobability sample space - Ω and pobability - P define a andom vaiable X oganize into the cumulative distibution function compute distibutional means and vaiances 3 / 11

4 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Definition of a Random Vaiable A andom vaiable is a eal valued function fom the pobability space. X : Ω R. Typically, we shall use capital lettes nea the end of the alphabet, e.g., X, Y, Z fo andom vaiables. The ange of a andom vaiable is called the state space. Execise. Give some andom vaiables on the following pobability spaces, Ω. 1. Roll a die 3 times and conside the sample space Ω = {(i, j, k); i, j, k = 1, 2, 3, 4, 5, 6}. 2. Flip a coin 10 times and conside the sample space Ω, the set of 10-tuples of heads and tails. We can ceate new andom vaiables via composition of functions: ω X (ω) g(x (ω)) Thus, if X is a andom vaiable, then so ae X 2, exp αx, X 2 + 1, tan 2 X, X 4 / 11

5 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Distibution Functions A (cumulative) distibution function of a andom vaiable X is defined by F X (x) = P{ω Ω; X (ω) x} = P{X x}. Fo the complement of {X x}, we have the suvival function F X (x) = P{X > x} = 1 P{X x} = 1 F X (x). Choose a < b, then the event {X a} {X b}. Thei set theoetic diffeence {X b} \ {X a} = {a < X b}. Consequently, by the diffeence ule fo pobabilities, P{a < X b} = P({X b} \ {X a}) = P{X b} P{X a} = F X (b) F X (a). In paticula, F X is non-deceasing. 5 / 11

6 and less than o equal to b. Consequently, by the diffeence ule fo pobabilities, Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions P {a <Xapple b} = P ({X apple b}\{x apple a}) =P {X apple b} P {X apple a} = F X(b) F X(a). (7.2) Distibution Functions Thus, we can compute the pobability that a andom vaiable takes values in an inteval by subtacting the distibution function evaluated at the endpoints of the intevals. Cae is needed on the issue of the inclusion o exclusion of the endpoints of the inteval. Let X be the sum of the values on two fai dice, Example 7.8. To give the cumulative distibution function fo X, the sum of the values fo two olls of a die, we stat with the table x x P{X = x} 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 P {X = x} 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 Hee is F X and the ceate cumulative the gaph. distibution function. 1 3/4 6 1/2 1/ Figue 7.1: Gaph of FX, the cumulative distibution function fo the sum of the values fo two olls of a die. 6 / 11

7 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Distibution Functions Notice that the distibution function is constant in between the possible values fo X, has a jump size at x is equal to P{X = x}, and is ight continuous. Call X a discete andom vaiable if its distibution function F X has these popeties. Examples = P{X = 4} = F X (4) F X (4 ) = P{4 < X 7} = F X (7) F X (4) = = = P{4 X 7} = F X (7) F X (4 ) = = = / 11

8 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Distibution Functions Execise. 1. Flip a fai coins 3 times. Let X be the numbe of heads. Unde equally likely outcomes, find P{X = x} fo x = 0, 1, 2, and 3. and use this to sketch a gaph of the distibution function F X. 2. Deal 5 cads out of a deck of 52. Let X be the numbe of. Unde equally likely outcomes, use the choose function in R to detemine P{X = x} fo x = 0, 1, 2, 3, 4, and 5. and use this to sketch a gaph of the distibution function F X. 8 / 11

9 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Distibution Functions Fo a dat boad with adius 1, assume that the dat lands andomly unifomly. Let X be the distance fom the cente. Fo x [0, 1], F X (x) = P{X x} = aea inside cicle of adius x aea of cicle = πx 2 π1 2 = x 2. Thus, we have the distibution function 0 if x 0, F X (x) = x 2 if 0 < x 1, 1 if x > 1. pobability !0.2!0.4!0.6!0.8! !1!0.8!0.6!0.4! x x 9 / 11

10 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Distibution Functions Execise. 1. Find the pobability that the dat no moe than 1/2 unit fom the cente. 2. Find the pobability that the dat lands futhe 1/3 unit but no moe than 2/3 unit fom the cente. 3. Find the median, x 1/2 so that P{X x 1/2 } = 1/2. Definition. X is continuous andom vaiable if it has a cumulative distibution function F X that is diffeentiable. 10 / 11

11 Definition of a Random Vaiable Distibution Functions Popeties of Distibution Functions Popeties of Distibution Functions A distibution function F X has the popety that it is ight continuous, stats at 0, ends at 1, and does not decease with inceasing values of x. In mathematical tems, Fo evey a, lim x a+ F X (x) = F X (a). lim x F X (x) = 0. lim x F X (x) = 1. Fo evey a, b satisfying a < b, F X (a) F x (b). distibution function x 11 / 11

DISTRIBUTION MIXTURES

Application Example 7 DISTRIBUTION MIXTURES One fequently deals with andom vaiables the distibution of which depends on vaious factos. One example is the distibution of atmospheic paametes such as wind