1 Econ 310 Professor Wallace October 1, 2013 Lecture: - Poisson distribution - Continuous probability distributions o Probability density function o Expected value and variance - Specific continuous probability distributions o Continuous uniform distribution o Normal distribution o Standard normal distribution and standardizing transformation2 The Poisson Probability Distribution The Poisson distribution describes the probability of that x number events occur in an interval of time or space. Illustration 1: The number of job offers that an unemployed worker receives in a week might have a Poisson distribution. Illustration 2: The number of defective parts that come off the assembly line per hour might also have a Poisson distribution Poisson random variable - the number of successes that occur in a period of time or an interval of space in a Poisson experiment Properties of a Poisson experiment 1. The number of events that occur in any interval is independent of the number of events that occur in any other interval 2. The probability of success in an interval is the same for all equal-size intervals 3. The probability of success in an interval is proportional to the size of the interval. 4. The probability of more than one success in an interval approaches 0 as the interval becomes small. The Poisson PF: ()!xefxx where ()Ex the expected number of events occurring in an interval of time or space and 2.71828e  $ The mean of a Poisson distributed random variable is always going to be a rate per unit of time or space.3 Example 1: An unemployed worker receives an average of 0.25 job offers per week. What is the chance that he receives 1 offer in one week?  10.250.25(1) (1) 0.19471!ePf   Example 2: An unemployed worker receives an average of 0.25 job offers per week. What is the chance that he receives 2 offers in one week? 2 0.250.25(2) (2) 0.0182!ePf   Example 3: An unemployed worker receives an average of 0.25 job offers per week. What is the chance that he receives 2 offers in 4 weeks? Because the probability is the same over any two intervals of equal length and the probability of the occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval you can always calculate the probability of x events in an interval by multiplying the mean be a factor of proportionality. Because 4 weeks is 4 times one week we can multiply the original Poisson mean of 0.25 per week by 4 to obtain a mean for a 4 week period. 211(2 offers in 4 weeks) (2) 0.1839! 2!xeePfx   4 Example 4: An unemployed worker receives an average of 0.25 job offers per week. What is the probability that he receives 10 offers in a year? 10 1313(10 offers in 52 weeks) (10) 0.0859! 10!xeePfx    Example 4: An unemployed worker receives an average of 0.25 job offers per week. What is the probability that he receives 11 or more offers in one year? 1310 100013( 11) 1 (10) 1 ( ) 1 0.7483!xxxeP x F f xx        You would probably want to use STATA or Excel for this calculation. Using STATA . display 1-poisson(13,10) 0.748317975 Continuous Probability Distributions Probability function (PF) versus probability density function (PDF) - Probability functions are applicable to discrete random variables whereas probability density functions are applicable to continuous random variables. - Probability functions indicate the probability that the random variable equals the value inputted into the function. The probability that a continuous random variable will take on any particular value is equal to zero. - The area under a probability density function over an interval is the probability of the random variable falling in that interval. Illustration1: If ()fx is the PDF associated with the random variable ,X then 211 2 1( ) ( ) ( )xxP x X x P x X x f x dx       Illustration 2: If ()fx is the PDF associated with the random variable X with range a x b, then the cumulative distribution function is ( ) ( ) ( )xaF x P X x f x dx   Requirements for probability density functions The following requirements apply to a probability density function ()fx with range a x b: 1. ( ) 0fx for all [ , ]x a b 2. The total area under ()fx in the interval [ , ]ab is 1.6 Expected Value and Variance of Continuous Random Variables Expected value: ( ) ( )XxSE X x f x dx Variance:  2( ) ( ) ( )XxSVar X x E x f x dx   - All previously discussed expected value, variance, and covariance properties apply to both discrete and continuous random variables.7 Continuous Uniform Probability Distribution Continuous Uniform PDF: 1( ) f x for a x bba   Continuous Uniform CDF: ( ) xaF x for a x bba   The distinguishing feature of the continuous uniform distribution is that the probability that a random variable falls in any two intervals of equal length is equal Example 1: Suppose that the PDF associated with a continuous random variable is ( ) 1/10 for 0 x 10fx   a. Is this random variable uniformly distributed? b. What are the values of a and b? c. What is the CDF? d. What does this PDF look like graphically? 108 e. What is probability of that X is less than 5 f. What is the probability that X is between 3 and 6? g. What is the probability X is greater than 8? h. What is the probability that X equals 4? $ For continuous random having the CDF is critical to computing probabilities.9 Example 2: Suppose that ~ ( , ).X U a b Calculate the mean and variance of .X  2 2 21( ) ( )2 2( )( )( ) 2( ) 2Xbbx S aax b aE X x f x dx x dxb a b a b aa b a b a bab           2222223223 3 2 2 2 2( ) ( ) ( ) ( )21 2 3( ) 2( )( ) 2 3( ) 3( ) 2 3( ) 4 XxSbbaaabVar X E X E X x f x dxa b x a bx dxb a b ab a a b b a b ab a b ab ab a b a b a                                 2 2 2 2 2 2 2 22 2 22 4( ) 3( 2 ) 3 4 12 122 ( )