Question

3. Two turtles are racing. The length of time that turtle A takes is expo 2 nentially distributed with mean 5 minutes. The length that turtle B take is also exponentially distributed but with mean 7 minutes. Assume tha their times are independent. (a) What is the probability that A wins? (b) What is the probability that the winner takes longer than 6 minutes (c) What is the expected time of the winner? (d) What is the probability of a tie?

Answer 1

$Let~X~be~random~variable~denoting~length~of~time~that~turtle~A~takes\\ Y~be~random~variable~denoting~length~of~time~that~turtle~B~takes\\ Then~X\sim exponential~with~mean=5\\ Y\sim exponential~with~mean=7\\ The~joint~distribution~of~X~and~Y~is\\ f_{X,Y}(x,y)=\frac{1}{35}\exp\left(-\frac{x}{5}-\frac{y}{7} \right ),~-\infty<x,y<\infty\\ =0~otherwise\\ (a)~Probability~that~A~wins=P(X<Y)=\frac{1}{35}\int_0^\infty\left(\int_0^y \exp\left(-\frac{x}{5} \right )dx\right )\exp\left(-\frac{y}{7} \right )dy\\ =\frac{1}{7}\int_0^\infty (1-\exp(-y/5))\exp(-y/7)dy=1-(1/7)*(35/12)=1-(5/12)=\frac{7}{12}\\ (b)~Probability~that~the~winner~takes~longer~than~6~minutes\\ =P(\min\{X,Y\}>6)=P(X>6,Y>6)=P(X>6)P(Y>6)\\ =\left( \frac{1}{7}\int_6^\infty \exp\left(-\frac{y}{7}\right)dy \right )\left( \frac{1}{5}\int_6^\infty \exp\left(-\frac{x}{5}\right)dx \right )=\exp\left(-6(1/7&plus;1/5) \right )\\ =0.1278\\$

$(c)~The~cdf~of~U=\min\{X,Y\}~is\\ F_U(u)=P(\min\{X,Y\}\leq u)=1-P(X>u,Y>u)\\ =1-\left( \frac{1}{7}\int_u^\infty \exp\left(-\frac{y}{7}\right)dy \right )\left( \frac{1}{5}\int_u^\infty \exp\left(-\frac{x}{5}\right)dx \right )\\ =1-\exp\left(-\frac{12u}{35} \right ),~u>0\\ The~pdf~of~U~is\\ f_U(u)=\frac{\mathrm{d} F_U(u)}{\mathrm{d} u}=\frac{12}{35}\exp\left(-\frac{12u}{35} \right ),~u>0\\ =0~otherwise\\ Hence~U\sim exponential~distribution~with~mean=\frac{35}{12}.\\ Hence~expected~time~of~the~winner=35/12=2.92~min.\\ (d)~P(X=Y)=\frac{1}{35}\int_0^\infty \exp\left(-\frac{12x}{35}\right)dx=\frac{1}{12}$