Attempts: Keep the Highest: 75 6. The exponential distribution An economist studied a large data set of Mexican consumer prices covering episodes of both high and low inflation. One of the goods in the study was toothpaste. When the inflation rate was low, an average of 1.9 changes in the price of toothpaste occurred each year. When the inflation rate was high, the price of toothpaste changed more frequently-an average of 8.4 times each year. (Source: E. Etienne Gagnon, Price setting during low and high inflation: Evidence from Mexico, International Finance Discussion Papers, No. 896 (City: Board of Governors of the Federal Reserve System, 2007)-1 The expected number of toothpaste-price changes in a 1-month period is when inflation is low and when inflation is high Assume that the number of price changes in any 1 month period, is described by a Poisson probability distribution with a mean equal to one of the values you just calculated (depending on whether the inflation rate is high or low). Then, the number of months between consecutive price changes is exponentially distributed with a mean of of when inflation is high when inflation is low and a mean How use the following Distributions tool to help you answer some of the questions that follow Select a Distribution Exponential Poisson Distributions when inflation is low and when inflation The probability that the price of toothpaste remains fixed for more than 3 months is to high. If the probability that the price of toothpaste stays the same for 2 months or less is about 0.76, is Mexico's inflation rate high or low? LOW High M o nahoton epsode, the variance of is

Solution - 1) Required to find the expected no.Of toothpaste- price changes in a 1-month period is when inflation is low & high : .. It can be founded by dividing the average no. of mille price changes in year for both when the inflation is low as well as cohen high when infection is loco - 19 = the I low = 0.1583 when inflation is high = 8. -840 120 [high = 0.7 ? Given x, the number of monks beheen conseccetive price clanges, i&scaron; esyonentia Ilg distributed with a mean of when in frantion is low & high :- mean I 01 W.K.E ypowo ro poisson (nowo = 0.1583 1 8 Shigh a passion ("high = 0.7) Now, we know that whenever yo passion (1) Then the inter event time bekeen events of y(say, XJ is exponentially distributed coith a posameeter and mean ut Xlow is exponentally distributed with a 02 0.1583 cm (X low = 63170 000, Ounsaction & Xheign is Exponentially distributed with a mean i ng isinabi niso 1 Frigh = 1.4285 3) The probability of price final ke semains 1. fixed more than 3 months is when inflation & - low & heigh:- The required probabilities are P ( Xpoco >3) & P(Xhigh >3).-0.7 Then , Bi l l 82') : -0.4749 Now, plXlad > 3) = 1- P(x200 (3) 0 -1-41-e Alcox3) se slow *3. Vir E using the formula for Cdf of exponential distribution) we get : P ( X loco >3) = è la 3! e-0.158343 V FP Xlco >3) = 0.6219) plxnegn >3)=e-1 high 3 Y6 10.03. -0.7x3 :: e-2-1 1 PC high >3) = 0-1224 4) The probabinity of the price fmille stays the same for 2 months (or) less is about 075 Ts mexicos infection and high con) low'. I plx.2)=076 , using the formula for the cdf of an Exponential distzibution , we get :- 1-eth = 0.76 é Ano 21 -0.76 = 0. 24 -) ->X2 - In (0.24) =) -4x2 = -1.4271 =) +de+1.4271 2 -0.713557 ..Which is approximately equal to low Thus mexico's infection rate is low s) If mexico is in a high inflation Episode the variance of, Xis, W.Kit Xneon ~ Exponertkal (nigh=0.7) Theis variance of Xhigh is :- var (Xhigh) = Carign? = 10.72 sata 0.49306). Con = 2:0408000 var(Wrign) = 2.04 .