# Poisson probability

Discussion in 'Math' started by boks, Feb 23, 2009.

1. ### boks Thread Starter Active Member

Oct 10, 2008
218
0
The number of ships X that arrives at a port during a day has proven to be given by a Poisson distribution with E(X) = 2. Which number of ships has largest probability of arriving on a certain day?

Attempt.

$
p(x;2) = \frac{e^{-2} \cdot 2^x}{x!}
$

The formula yields

p(0;2) = 0.135
p(1;2) = 0.270
p(2;2) = 0.270
p(3;2) = 0.180

and then values of p approaching zero. The numbers of ships most likely to arrive are 1 or 2.

The port can handle at most 3 ships a day. Redundant ships are sent to another port.
What's the expected number of ships that are handled each day?

Expected number= 1*0.27 + 2*0.27 + 3* 0.18 = 1.35 (Correct: 1.782)

Which capacity is necassary to be able to handle all incoming ships on a day with 90% certainty?

No idea...

Last edited: Feb 23, 2009
2. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
You have three questions here by my count. Here are some suggestions for answering them.
1. "Which number of ships has largest probability of arriving on a certain day?"
For what value of x (not limited to integer values) is p(x, 2) the largest? To do this, you would need to differentiate p(x, 2) set p'(x, 2) to zero, and solve for x. A complication is that p(x, 2) has a factorial in the denominator, but you can replace x! with Gamma(x + 1).

2. "The port can handle at most 3 ships a day. Redundant ships are sent to another port. What's the expected number of ships that are handled each day?"
This might be a naive approach, but isn't the expected number of ships E(x)? For a Poisson distribution, the mean and variance = lambda = E(x), which you give as 2.

3. "Which capacity is necassary to be able to handle all incoming ships on a day with 90% certainty?"
I think what you want here is the value of x for which p(x, 2) <= .9, which you should be able to find from a table of values for Poisson distributions. The idea is that you want to find the value of x for which 90% of the area under the curve lies to the left of x, and 10% of the area lies to the right.