I've got this lengthy question that I'm really struggling with, I'd appreciate all help i can get.
"The number of customers Y arriving at a walk-in shop in the first t minutes after it opens doors [i.e time interval (0,t)] on any particular day follows a poisson distribution with mean $\lambda t$.
[the parameter $\lambda$ is the rate of arrivals per unit time and $t$ is the length of time period, so $\lambda t$ represents the mean of number of customers arriving in the interval of length $t$.
a) Let the random variable $T$ denote the time (in minutes) until the arrival of the first customer. Determine the probability density function of $T$.
[Hint: if $T>a$, then no customers have arrived in the time period (0,a).]
b) Let the random variable $U$ be the time until the arrival of the second customer. Show that $U$ has a gamma distribution with $\alpha = 2$ and $\beta=1/\lambda$
c) Let the random variable $W$ be the time until the arrival of the $k$th customer. Show that $W$ has a gamma distribution with $\alpha=k$ and $\beta=1/\lambda$.
I'm assuming that for part a, i need to solve $P(Y \le 1)=P(Y=0)+P(Y=1)$ for
$P(Y=y)=$ $(\lambda t)^y e^{-\lambda t} \over y!$
and then solve for $t$ from there, but i'm not really sure.
as for part b and c, I'm not really sure, as they both seem to depend on the solving the first part.
I know it's a lengthy question, and it'd take some time to get it solved.
All the help is appreciated.
Cheers
As part $a)$ was addressed in the comments and $b)$ is a special case of $c)$ which you can verify after seeing the general idea, I only present hints for $c)$.
Again, note that if $T_j$ is the time until the $j$-th customer arrives, then $T_j >t $ if and only if $N_t < j$. Thus, $P(T_j >t)=P(N_t \leq j-1)$. Writing as before, we have $F_{T_j}(t)=1-P(N_t \leq j-1)$. But this is equal to $$1-e^{-\lambda t}\left(1+\lambda t+\frac1{2!}(\lambda t)^2+\dotsc +\frac1{(j-1)!}(\lambda t)^{j-1}\right).$$
Differentiate, with respect to $t$, the above expression to get $f_{T_j}(t)=\frac{\lambda^j}{(j-1)!}t^{j-1}e^{-\lambda t}$ and recognize this is the PDF of r.v. with Gamma distribution, $\frac{1}{\Gamma(\alpha)\beta^\alpha} t^{\alpha-1}e^{- t/\beta}$, where $\alpha=j$, and $\beta=1/\lambda$, QED.