Probability Distributions

Transcription

1 9//5 : (Discrete) Random Variables For a given sample space S of some experiment, a random variable (RV) is any rule that associates a number with each outcome of S. In mathematical language, a random variable is a function whose domain is the sample space and whose range is the set of real numbers. To put it simply, a random variable is a variable that takes on numerical values that depend on the outcome of a chance operation. In we will discuss discrete random variables. In CH4 we will work with continuous random variables. Probability Distributions The probability distribution or probability mass function (pmf) of a discrete RV is defined for every number x by p(=p(x=. For every possible value x of the RV, the pmf specifies the probability of observing that value when the experiment is performed. The conditions: p( p(= are required of any pmf. 4

2 9//5 Example. Consider selecting at random a student who is among the 5, registered for the current term at Mega University. Let X= the number of classes for which the selected student is registered, and suppose X has the following pmf: x p( We can summarize the distribution graphically: p( x = # Classes 5 6 What is the probability that a randomly selected student will be registered for 4 classes? Find P( X 5) Cumulative Distribution Function The cumulative distribution function (cdf) F( of a discrete RV variable X with pmf p( is defined for every member x by F( P( X For any member x, F( is the probability that the observed value of X will be at most x. y: yx p( y) 7 8

3 9//5 Example. (continued) The pmf of X (number of classes) is: x p( Then the cdf is: F(= if x<. if x<.4 if x<.7 if x<4.4 if 4 x<5.8 if 5 x<6.98 if 6 x<7. if 7 x Expectation of X Let X be a discrete RV with pmf p(. The expectation or mean value of X, denoted by E(X) or µ X is E( X ) x p( X 9 Example. (continued) Since the total number of enrolled students is 5,, we can find the number of students registered for a given number of classes: x p( # The mean (or average) # of classes per student: Expectation of a Function If the RV X has a pmf p(, then the expectation of any function h(x) denoted E[h(X)] or µ h(x) is computed by: E[ h( X )] h( p( The expected value for a linear function follows directly: E( ax b) a E( X ) b

4 9//5 Variance of X Let X have pmf p( and expectation µ. Then the variance of X, denoted V(X) or σ is: V ( X ) ( x ) p( E[( X ) ] Shortcut Formula for σ V ( X ) [ x p( ] E( X ) [ E( X )] The standard deviation (SD) of X is: 4 Example. (continued) x p( V ( X ) ( x ) p( Using the Shortcut formula. E( X ) V ( X ) E( X ) [ E( X )].4 The Binomial Probability Distribution Binomial Experiments: The experiment consists of a sequence of n smaller experiments called trials where n is fixed in advance of the experiment. Each trial can result in one of the same two possible outcomes ( Success or Failure ) The trials are independent, so that the outcome of any particular trial does not influence the outcome of any other trial. The probability of success is constant from trial to trial; we denote this probability p

5 9//5 Example: Imagine flipping a coin. We will call Heads a success and Tails a failure. Assuming the coin is fair, then p=.5 (Heads and Tails are equally likely). In two tosses (n=), we have already determined the probability of observing, or Heads (successes) using a probability tree: Number of Heads Probability This is a binomial distribution with p=.5 and n=. Binomial Random Variable X The binomial random variable X associated with a binomial experiment consisting of n trials is defined as X= the number of successes among the n trials Because the pmf of a binomial RV X depends on the two parameters n and p, we denote the pmf by Bin(x;n,p). 7 8 Binomial Probability Formula Binomial Distributions with p=.5 and various values of n Bin( x; n, p) n p x x nx ( ) ( ) P X x p Probability Binomial Distribution with p=.5 and n= Probability.4.. Binomial Distribution with p=.5 and n=.... Binomial Distribution with p=.5 and n=5 Binomial Distribution with p=.5 and n= Probability..5 Probability

6 9//5 Binomial Distributions with n= and various values of p Example: Binomial Distribution with n=,p=.5 P( X ) p ( p).5 (.5).5.5 P( X ) p ( p).5 (.5) P( X ) p ( p).5 (.5) P( X ) p ( p).5 (.5).5.5 Probability Probability.4... Binomial Distribution with p=.5 and n=. Expectation and Variance of a Binomial RV If X~Bin(n,p), then E( X ) np V ( X ) np( p) np( p) Example: Circuit Boards When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let X=the number of defective boards in a random sample of size 5, so X~Bin(5,.5). x P(X= P(X E E E E-9..7E-. 5.4E-. 9.5E-4..6E E E-. 4 6

7 9//5 What is the probability that none of the 5 circuit boards is defective? Find E(X) Determine P(X ) Find V(X) Determine P(X 5) Poisson Probability Distribution A random variable X is said to have a Poisson distribution with parameter λ (λ>) if the pmf of X is x e p( x, ) P( X for x,,, x! The value of λ is frequently a rate per unit time or per unit area. The letter e represents the base of the natural logarithm: e=.788. Expectation and Variance of a Poisson RV If X has a Poisson distribution with parameter λ, then E(X)=V(X)= λ

8 9//5 Example.9: Clams Let X denote the number of clams captured in a trap during a given time period. Suppose that X has a Poisson distribution with λ=4.5, so on average traps will contain 4.5 clams. x P(X= P(X Probability Mass Poisson Distribution: Mean = x 9 Find the probability that a trap contains exactly 5 clams. Find the probability that a trap contains at most 5 clams. Relationship between Poisson and Binomial Distributions For binomial experiments where n is large and p is small, the distribution is approximately Poisson with λ=np. As a rule of thumb, the approximation can be safely applied if n>5 and np<5. 8

9 9//5 Example: Glass Manufacturing Suppose out of windows have bubbles. From a batch of windows, what is the probability that fewer than will have bubbles? Binomial distribution with n=, p=.. Is the Poisson Approximation reasonable here? If so, what is the value of λ? 4 Bin(n=,p=.) x P(X Probability Mass Binomial Distribution: Trials =, Probability of success = x P(X Poisson(λ=) Probability Mass Poisson Distribution: Mean = x 5 6 9

10 9//5 Poisson Process If the number of events that can occur in a time interval are independent with a mean rate λ and there are t disjoint time intervals, then X=the number of events occurring in the t time intervals follows a Poisson distribution with mean λt. Example: Cars There is an intersection that, during the night, will average 5 cars per minute approaching it. What is the probability that exactly 8 cars reach the intersection during a three minute period. 7 8 Key words Probability mass function (pmf) Cumulative distribution function (cdf) Expectation Variance and standard deviation Binomial random variable Poisson random variable 9