I understand pretty well the concept of Bernoulli experiment and binomial distribution. However today I faced a question like this one.
A traffic engineer is studying a left-turn lane that is long enough to hold $7$ cars. Let $X$ be the number of cars in the lane at the end of a randomly chosen red light. The traffic engineer believes that the probability that $X=x$ is proportional to $(x+1)(8-x)$ and $x= 0,1, \dots 7.$
Of course, the first thing I wanted to do was set up a PMF that allowed me to see the probability, however, I was confused by "proportional".
Can someone clarify this idea please?
Any help is greatly appriciated.
EDIT: hold one, now I am wondering, is it even binomial?
The distribution is certainly not binomial. That the pmf is proportional to the given expression means that $P(X=x)=k(x+1)(8-x)$, where $k$ is such that $P(X=0)+\dots+P(X=7)=1$ to satisft the requirement of a probability distribution to sum to 1 over all events. $k=\frac1{120}$ here.