Assign 2_Random_var

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

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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
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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
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3 The Geometric Random Variable
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Assign 2_Random_var - 2 2.1 Random Variables Random...

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