$($m$)$ An investor has purchased the bottom tranche of a mortgage$-$backed security $($MBS$)$ con$-$

sisting of $10$ similar mortgages. This is the tranche that is first affected when one of the

mortgages that make up the MBS defaults, and thus the investor is particularly interested

in the first time $T$ that one of the mortgages defaults.

As a simplifying assumption, the investor assumes that the default times ${}_1,$dots,${}_{10}$ of the

mortgages that make up the MBS are independent exponentially distributed with parameter

$.$ Then the first$-$to$-$default time is $T$:$=$min$\{{}_1,$dots,${}_{10}\}.$

By looking at historical data, the investor notes that five MBS with identical structure and

characteristics had first$-$to$-$default times

$T_1=1\mathrm{.}2,T_2=0\mathrm{.}9,T_3=2\mathrm{.}1,T_4=1\mathrm{.}5,T_5=1\mathrm{.}3.$

What is the maximum likelihood estimate estimate of $?$ Round your answer to $3$ decimal

places.

$[$Hint: First compute the PDF of $T.]$