Assign 2_Random_var

# Assign 2_Random_var - 2 2.1 Random Variables Random...

This preview shows pages 1–4. Sign up to view the full content.

2 Random Variables 2.1 Random variables Real valued-functions deFned on the sample space, are known as random variables (r.v.’s): RV : S R Example. X is a randomly selected number from a set of 1, 2, 4, 5, 6, 10. Y is the number heads that has occured in tossing a coin 10 times. V is the height of a randomly selected student. U is a randomly selected number from the interval (0 , 1). Discrete and Continuous Random Variables Random variables may take either a Fnite or a countable number of possible values. Such random variables are called discrete . However, there also exist random variables that take on a continuum of possible values. These are known as continuous random variables. Example Let X be the number of tosses needed to get the Frst head. Example Let U be a number randomly selected from the interval [0,1]. Distribution Function The cumulative distribution function (c.d.f) (or simply the distribution function ) of the random variable X ,sayit F , is a function deFned by F ( x )= P ( X x ) x R . Here are some properties of the c.d.f F , (i) F ( x ) is a nondecreasing function, (ii) lim x →∞ F ( x )=1 (iii) lim x →-∞ F ( x )=0 All probability questions about X can be answered in terms of the c.d.f F . ±or instance, P ( a<X b F ( b ) - F ( a ) If we desire the probability that X is strictly smaller than b , we may calculate this probability by P ( X<b ) = lim h 0 + P ( X b - h ) = lim h 0 + F ( b - h ) Remark. Note that P ( ) does not necessarily equal F ( b ). 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 2.2 Discrete Random Variables Defnition. (Discrete Random Variable) A random variable that can take on at most a countable number of possible values is said to be discrete . For a discrete random variable X , we defne the probability mass function (or probability density ±unction, p.d.±) o± X by p ( a )= P ( X = a ) . Let X be a random variable takes the values x 1 ,x 2 ,... . Then we must have ± i =1 p ( x i )=1 . The distribution ±unction F can be expressed in terms o± the mass ±unction by F ( a ± all x i a p ( x i ) Example. Let X be a number randomly selected ±rom the set o± numbers 0, 1, 2, 3, 4, 5. Find the probability that P ( X 4). The Binomial Random Variable Suppose that n independent trials, each o± which results in a ”success” with probability p and in a ”±ailure” with probability 1 - p , are to be per±ormed. I± X represents the number o± successes that occur in the n trials, then X is said to be a binomial random variable with parameters ( n,p ). Denote X B ( ). The probability mass ±unction o± a binomial random variable with parameters ( )i s given by P ( X = k p ( k ² n k ³ p k (1 - p ) n - k ,k =0 , 1 , 2 ,...,n where ² n k ³ = n ! k !( n - k )! . Note that n ± k =0 p ( k n ± k =0 ² n k ³ p k (1 - p ) n - k =( p +(1 - p )) n =1 Example. According to a CNN/USA Today poll, approximately 70% o± Americans believe the IRS abuses its power. Let X equal the number o± people who believe the IRS abuses its power in a random sample o± n=20 Americans. Assuming that the poll results still valid, fnd the probability that (a) X is at least 13 (b) X is at most 11
3 The Geometric Random Variable

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 02/23/2011 for the course MATH 444 taught by Professor Any during the Fall '10 term at Roosevelt.

### Page1 / 13

Assign 2_Random_var - 2 2.1 Random Variables Random...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online