Question

Statistics question using Poisson Probability Distribution:

Answer 1

"Time headway" in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let X = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow. The pdf of X is the following.

\[f(x)=\left\{\begin{matrix} 0.14e^{-0.14(x-0.5)} & x\geq 0.5\\ 0 & \textrm{Otherwise} \end{matrix}\right.\]
What is the probability that time headway is more than 4 sec? At least 4 sec?

I was able to find a). by integrating:
\[P(x>4)=1-\int\limits_{0.5}^{4}f(x)dx=0.613\]But apparently the answer is the same for b) and I cannot understand why =/

Answer 2

integration is a limiting process ... so the inclusion or exclusion of 4 amounts to the same value.

but isnt the poisson a discrete distribution?

Answer 3

Oh you're right, it's not discrete, not sure why I thought of putting that, but the calculation is correct.

How do you mean limiting? What I mean is why is it that \(P(X>4)=P(X \ge 4)\)?

Answer 4

For any continuous distribution, \(P(X=k)=0\), for pretty much the same reason that \(\displaystyle \int_c^cf(x)\,dx=0\) for some constant \(c\).

Answer 5

if we take the area as 'slices' ... as the slices tend to zero we get the area under the curve.

inclusion of, or approach to, a slice at 4 is essentially adding or leaving out an addition of 0

Answer 6

Ahhh that makes sense. I didn't think about it graphically, but that helps!! Thank you! :)

Answer 7

not sure how rigorous my thought is, but its in my head lol

Answer 8

discrete:

P(x>=4) = 1 - (P(0)+P(1)+P(2)+P(3))
P(x>4) = 1 - (P(0)+P(1)+P(2)+P(3)+P(4))