Answer:
Check the explanation
Step-by-step explanation:
Number of transactions in a day is sum of number of withdrawals and number of deposits. So,
Number of transactions in a day, Z = X + Y
Moment Generating function of Z is,
T+1
Expected number of transactions in a day = E[Z]
[tex]= \frac{\mathrm{d} }{\mathrm{d} t}_{t=0}M_Z(t) = (r+1)\left ( \frac{p}{1-qe^t} \right )^{r} * \frac{-p}{(1-qe^t)^2} * (-qe^t) for t = 0[/tex]
[tex]= (r+1)\left ( \frac{p}{1-qe^0} \right )^{r} * \frac{-p}{(1-qe^0)^2} * (-qe^0)[/tex]
[tex]= (r+1)\left ( \frac{p}{1-q} \right )^{r} * \frac{-p}{(1-q)^2} * (-q)[/tex]
[tex]= (r+1)\left ( \frac{p}{p} \right )^{r} * \frac{-p}{(p)^2} * (-q)[/tex]
[tex]= \frac{(r+1)q}{p}[/tex]
