$[$Calvin tries to call Hobbes$]$

Calvin is trying to call Hobbes but he doesn't know if the number he's calling is his home phone or his work phone. The probability that Hobbes answers at his home phone is $p_0,$ whereas the probability that Hobbes answers at his work phone is $p_1,$ with $xxGeometric(p_i)H_{i}2{}_0=3{}_1{H}_1{p}_{MD},p_{FA}{p}_e{p}_0=e^{-1},p_1=1-e^{-1}{H}_1{p}_0{p}_1{p}_{MD},p_{FA}{p}_e{p}_0=e^{-1},p_1=1-e^{-1}0.$ Calvin calls the number $he$ has once a day until Hobbes finally picks $up.$ Let $x$ denote the number $of$ times Calvin has $to$ call $up$ until Hobbes finally answers $(the$ answered call included $in$ the count$).$ Assume that the successive call attempts are independent trials $ofan$ experiment, $so$ that $xGeometric(p_i),$ under hypothesis $H_i.$ Assume that $2{}_0=3{}_1.$

$(a)$ Obtain the $ML$ decision rule. Assume ties are broken $in$ favor $of{H}_1.$

$(b)$ Obtain $p_{MD},p_{FA},$ and $p_e$ under the $ML$ decision rule assuming $p_0=e^{-1},p_1=1-e^{-1}.$

$(c)$ Obtain the MAP decision rule. Assume ties are broken $in$ favor $of{H}_1,$ and leave $p_0$ and $p_{1}as$ variables $(do$ not use the values given $in$ part $(b)).$

$(d)$ Obtain $p_{MD},p_{FA},$ and $p_e$ under the MAP decision rule assuming $p_0=e^{-1},p_1=1-e^{-1}.$