University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

113

Pages:

3
Author:

Niklas

LECTURE 19 Cumulative distribution function (discrete rv). Given a random variable X that assumes a countably infinite set of values X, X, Xn... and with probability mass function p(x), we define the cumulative distribution function X as: F(x)=P(x)=Σ ρίχιζ The cumulative distribution function represent the probability that X does not exceed the value x 1. FIXIPO VER ii. F(x) is non decreasing iii. lim F(x)=0 iv lim F(x)-1 Uniform disaete distribution A raudom variable xefa, a+1, 2+2,.., 6-2, 6-1, b) follows a discrete uniform distribution on the interval [a,b] iR and only if its probability mass function is: P(xx)= where a and b are integer numbers such that asb and 1=b-a+1 - X~ Unif [a,b] X all the values of the support are equally likely to be observed -The expected value and variance of a RV X following a discrete uniform distribution are respectively Exampur rolling a dice a+b E(x)2 V(x)= 12 2 3 S 6 x={outcome dice]=[1,2,3,4,5,6] P(xx) meau: E(x)= a+b 4+6 2 variance (x)= 3.5 -462-4 42 2.92 P(x3)=P(x)=-50% Geometric distribu The raudone variable X describing the number of trials needed before the first success follows a geometric distribution with probability mass function where ocps4 is the probability of success -The support of the random variable X is 4,2,3, -X~ Geo(p) is the number of failures before the first success in independent and equally likely trials with probability of success equal to P -The expected value and variance of a rv X following a Geometric distribution are respectively E(x)=(x)=- 4-P pa Example tossing a min Ti failure 1-P H: successP Tasting the coin 5 times, Zing the probability that the last time you told you get a head TTTTH P(x+5)*(1-p)(4-p)(1-p) (1-P) P=(4-p) p Hypergsometric distribution From a set N objects of which I have a certain characteristic, n balis/cards are expected without X=#h-type objects on n extracted P(xx) for x au integer number and d(st) ()() (2) x3 max(0n+H-n) *ksmin (n.) otherwise the probability is zero Example soss a coin If we flip a coin three times we cau define several ru such as: S:#tails Hi tails before the first head Outcomes P S H SP 1 8 SE{0,1,2,3 0 1 HHT THT 4 THH 3 1 8 HTT 2 1 8 TTH 2 0423 HHT THT ト TTT 33 4 2 TTH 3 TTT We want to look at them jointly, so we consider events as: (Sis) (Hom) HP не{0,1,2,3] T 31 0423 1/82/8/8 8/8 2 18 3 margines 8 1/8 / 8/8 1/8 3/8 3/8 /8 1 marginal tich Jaint probability distribution Let X and Y be pair of discrete variables with values (x1,...) and {y}. Their joint probabiliky distribution expresses the probability that simultaneously x takes the specific value xi and Y takes the value y; P(x, y;) P(x=x: Yzy) It is easy to see the link with the probability of the intersection of bivariats events Properties: - p(x,y) ≥0 Σ=1 = P(x,y) = 1 In this framework, the probability distribution of the raudom variable X is called its marginal probability. distribution and is obtained by summing the joint probabilities over all possible values, that is px(x) = P(Xaxi) = p(xi,j)