Contagious Distributions Stephen Nicar ECON7818 Midterm Question 6 Consider the data generating process wi Gi i where wi is the weight of individual i Gi 1 if the individual is a male and zero otherwise and each i is an independent draw from a normal distribution with mean 0 and variance 2 Assume that 60 of the population is male and 40 female maybe the PhD program in Econ at C U Write down the density function for w f w in this population The density function for w is what s called a contagious distribution or a mixture From MGB p 122 if you have 0 x f1 x f2 x of density functions and a sequence p0 p1 p2 such P a sequence fP that pi 0 and pi 1 then pi fi x is a density function When G 0 wi i When G 1 wi i In either case w is just a function of a normally distributed random variable plus a constant or We know that N 0 2 So when G 0 w N 2 When G 1 w N 2 G 0 and G 1 occur with probabilities p0 4 and p1 6 respectively So we have f0 2 1 1 e 2 w 0 2 0 f1 2 1 1 e 2 w 1 2 1 p0 4 p1 6 As a result the distribution of w is a mixture of two normal distributions Specifically the distribution of w is the probability weighted average of the two different normal distributions that occur when G 0 and G 1 2 2 1 1 1 1 fW w 4 e 2 w 0 6 e 2 w 1 2 0 2 1 This distribution has five parameters mean weight for females difference in mean weight of males and females 02 variance for distribution of female weight 12 variance for distribution of male weight and p0 percentage of females in the population Because of how we specified w 02 and 12 will be equal 1 Suppose the mean weight for females is 110 and the mean weight for males is 160 each with a standard deviation of 10 Then our plot with 110 50 0 1 10 p0 4 looks like Because this is a continuous distribution if you randomly sample one individual from the population the probability that they weigh exactly 140 pounds is zero The probability that they weigh between 120 and 140 pounds is Z 140 2 2 1 1 1 1 4 e 2 w 110 10 6 e 2 w 160 10 dw 0 0765532 2 10 2 10 120 2 By varying the parameters you can end up with distributions that have very different shapes Plot with 110 40 0 15 1 20 p0 4 Z 140 4 120 2 2 1 1 1 1 e 2 w 110 15 6 e 2 w 150 20 2 15 2 20 dw 0 236935 Plot with 110 40 0 15 1 20 p0 5 Z 140 120 5 2 2 1 1 1 1 e 2 w 110 15 5 e 2 w 150 20 2 15 2 20 3 dw 0 235736 Plot with 110 50 0 15 1 5 p0 4 Z 140 120 4 2 2 1 1 1 1 e 2 w 110 15 6 e 2 w 160 5 2 15 2 5 4 dw 0 091916