STAT 6201 - Mathematical Statistics : Chapter 3 Random variables . 5 Event Prc . ) , , Random variables and distributions Let S be the sample space associated with a probability experiment. Assume that we have defined the events of interest and a probability assignment for each event. A random variable is a real-valued function that is defined on the sample space S. Examples: I X = number of H in 10 tosses is a random variable; I Y = 10 X = number of T in 10 tosses is a random variable; − I T = the temperature in this room; I Z = the price of Apple, Inc. stock tomorrow; I etc. The value of a random variable is not known before performing the probability experiment ! Range of values for a random variable I While we don’t know the value of a random variable ahead of time (before performing the experiment), we can (in most cases) list the possible range of values for a random variable. Examples : I X = number of H in 10 tosses is a random variable: X 0, 1, 2,...,10 2 { } I Y = 10 X = number of T in 10 tosses is a random variable Y 0,−1, 2,...,10 2 { } I N = number of coin flips until H is observed: N 1, 2, 3,... 2 { } I T = the temperature in this room; T [20.0, 90.0] 2 I Z = the price of Apple, Inc. stock tomorrow Z [0.0, ) 2 1 Discrete vs. conitnuous random variables I A random variable X is called a discrete if X can take only a finite or countable set of values X x , x ,...,x or X x , x , x ,... 2 { 1 2 n} 2 { 1 2 3 } I A random variable X is called continuous if it takes values in a subinterval of the real line, or the entire real line. X [a, b]orX (a, )orX R 2 2 1 2 a XE too , ] XE ( , a) Examples Classify the following variables as discrete or continuous: I X = number of H in 10 tosses is a random variable: X 0, 1, 2,...,10 discrete 2 { } I Y = 10 X = number of T in 10 tosses is a random variable Y 0,−1, 2,...,10 discrete 2 { } I N = number of coin flips until H is observed: N 1, 2, 3,... discrete . 2 { } I T = the temperature in this room; T [20.0, 90.0] 2 I Z = the price of Apple, Inc. stock tomorrow Z [0.0, ) 2 1 s:. Discrete random variables and their distributions Recall that a random variable X is called discrete if the range of possible values is finite or countably infinite. Without loss of generality, we will assume that X x , x , x ,... 2 { 1 2 3 } Since we can’t know the value of X before performing the experiment, we characterize X by describing how likely each value is. That is, for each value xi in the range of X ,wedefine . it . f (xi )=Pr(X = xi ) , 2,3 , which is the probability that X will take on the value xi . The collection of probabilities f (x1), f (x2), f (x3),... define the probability mass function (pmf) of X and they describe the distribution of X. Example Toss a coin three times and let X be the number of H observed. Find the probability mass function (pmf) of X . - 3 XELO , 1,2 , } " " 4 K 22 3 , b % % 18 Ho )=Pr(X=o ) = f ⇐ f4)=Pr(x=D = § fH=Pr(X=y= f fbtprlx =D = ft Example Roll two dice and let X be the sum of the two numbers. Find the pmf of X . XER , 3,4 , , ,l2 } fH=Pr(X=2)=3t 436256 #. " .fB)=Pr(X=D=}z fl4I=PrlX=4)= ... until observe H flip a coin you = coin . N # of Clips Find the pmf . of N Nell , 2,3 , . ... } f(D=Pr(N=D= E fH=Pr( Not =L . I = at 9 %D= Pr(N=2o)=#' . } : i w1=Pr(N=w)=#n Properties of the pmf Let X be a discrete random variable X x1, x2, x3,... with probability mass function 2 { } f (x1), f (x2), f (x3),... The numbers f (xi ), i =1, 2,... must satisfy the following properties I f (x ) 0; i ≥ 1 I f (x )+f (x )+f (x )+ = f (x )=1 1 2 3 ··· i Xi=1 Oefki ) < 1 Major discrete distributions Bernoulli distribution A discrete random variable X has a Bernoulli distribution if it can only take two possible values, sometimes X 0, 1 2 { } " " 1 = success " ' ' 0 = failure The pmf of a Bernoulli random variable is f (0) = Pr(X = 0) f (1) = Pr(X = 1) Convention: f (1) = Pr(X = 1) = p thus f (0) = Pr(X = 0) = 1 p. − Notation X Bernoulli(p). ⇠ Binomial distribution Consider an experiment whose outcome can be either success or failure. Assume that Pr(success) = p on each trial. Repeat this experiment n times. Examples: I Flip a coin 10 times, think of H as success and T as failure on each trial; I Observe 20 items coming out of a production line and record whether they are defective or not. I Perform a clinical trial and record whether a patient recovers from her symptoms or not. A random variable X has a Binomial distribution if it describes the total number of successes in n independent and identical trials. X Binomial(n, p) ⇠ Binomial distribution Repeat the same trial n times and let p be the probability of success on any given trial. Let X denote the total number of successes in n trials. We say that X Binomial(n, p) ( X has a Binomial distribution with parameters n, p).⇠ Note that X 0, 1, 2, 3,...,n 1, n 2 { − } What is the probability mass function (pmf) of X ? flu --Pr(X=o)=kpT ftp.prl#D=wp(rpymfH=Pr(Xz2)=(Y)p2ltpT2 " Ye k=o 1,2 ... flk)=Pr(X=k)=( )p4rA , , ,w Binomial distribution X Binomial(n, p) then ⇠ X 0, 1, 2,...,n 2 { } and X has probability mass function n i n i f (i)=Pr(X = i)= p (1 p) − i =0, 1, 2,...,n i − ✓ ◆ Binomial distribution Connection to the Bernoulli(p) distribution: Let p (0, 1) and consider n random variables X1, X2,...,Xn which are independent2 and identically distributed (iid) random variables with X Bernoulli(p) i ⇠ Then Y = X + X + + X 1 2 ··· n counts the number of “successes” among the n variates X1, X2,...,Xn. Thus, Y Binomial(n, p) ⇠ Exercise Suppose that a random variable X has the Binomial distribution with parameters n = 15 and p =0.5.FindPr(X < 6). 15 . XELO ,1 , , ) flitprlxai ) .fi?)piCi.pj5iPr(xc6)=Pr(X=o)tMX=Dt...tprlx=5)=lYHsilD'stCi9tHl.H4t.. Exercise Suppose that a random variable X has the Binomial distribution with parameters n =8and p =0.7.FindPr(X 5). ≥ exercise.