Q3
Discrete Random Variables
Expectation and Variance via Combinatorial Counting
View
Consider a particle moving on the coordinate plane, and denote the location of the particle at time $t \in \{ 0,1,2 , \ldots \}$ by $\left( X _ { t } , Y _ { t } \right)$. The initial location of the particle is $\left( X _ { 0 } , Y _ { 0 } \right) = ( 0,0 )$. Also, if $\left( X _ { t } , Y _ { t } \right) = ( a , b )$, then $\left( X _ { t + 1 } , Y _ { t + 1 } \right) = ( a + 1 , b )$ with probability $p , \left( X _ { t + 1 } , Y _ { t + 1 } \right) = ( a , b + 1 )$ with probability $q$, and $\left( X _ { t + 1 } , Y _ { t + 1 } \right) = ( a , b )$ with probability $1 - p - q$. Here, it is assumed that $p , q > 0 , p + q < 1$, and the movements of the particle at different time points are independent. Let $( X , Y )$ denote the location of the particle such that $\left( X _ { t + 1 } , Y _ { t + 1 } \right) = \left( X _ { t } , Y _ { t } \right)$ for the first time. Answer the following questions.
(1) Show that the probability that $( X , Y ) = ( 1,2 )$ is $3 p q ^ { 2 } ( 1 - p - q )$.
(2) For non-negative integers $n$, find the probability that $X + Y = n$.
(3) For non-negative integers $n$, let $f _ { n }$ denote the probability that $X = n$.
(a) Find $f _ { 0 }$.
(b) Express the probability that $X \geq n + 1$ given the condition $X \geq n$, using $f _ { 0 }$.
(c) Show that $f _ { n } = \left( 1 - f _ { 0 } \right) ^ { n } f _ { 0 }$.
(4) Express the expectation of $X$ using $p$ and $q$.
(5) Express the correlation coefficient between $X$ and $Y$ $$\frac { E \left[ \left( X - \mu _ { X } \right) \left( Y - \mu _ { Y } \right) \right] } { \sqrt { E \left[ \left( X - \mu _ { X } \right) ^ { 2 } \right] E \left[ \left( Y - \mu _ { Y } \right) ^ { 2 } \right] } }$$ using $p$ and $q$, where $\mu _ { X } = E [ X ]$ denotes the expectation of $X$ and $\mu _ { Y } = E [ Y ]$ denotes the expectation of $Y$.