How to Calculate Infinite Series for Poisson Distribution?

[kingwinner]

Homework Statement

Evaluate

∞
∑ [(e^-15 15^x) / x!]
x=16

15
∑ [(e^-15 15^x) / x!]
x=0

Homework Equations

The Attempt at a Solution

The only way I can think of is writing out every term explicitly and adding them on a calculator.
Is there any faster way (without having to write out every term explicitly) to calculate the above sums?


Thanks for any help!

 

[Dick]

The sum of the two series is e^(-15)*e^(15), right? If you want the sums individually you do need a calculator. If you want the total, it's pretty obvious.

 

[Mark44]

Homework Statement

Evaluate

∞
∑ [(e^-15 15^x) / x!]
x=16

15
∑ [(e^-15 15^x) / x!]
x=0

Homework Equations

The Attempt at a Solution

The only way I can think of is writing out every term explicitly and adding them on a calculator.
Is there any faster way (without having to write out every term explicitly) to calculate the above sums?


Thanks for any help!

∞
∑ [(e^-15 15^x) / x!]
x=0

= e^-15 *

∞
∑ (15^x) / x!
x=0

= 1, if that's any help.

 

[HallsofIvy]

Once you know the sum of the two series, since the first is finite, it's not all that hard to find the sum of 15^x/x! for x from 0 to 15 by hand and then get the other sum by subtracting. The only place you really NEED a calculator (though I would recommend it for the tedious multiplications, divisions, and subtractions) is to evaluate e^-15

 

[statdad]

Another approach - both of these relate to the Poisson distribution:

$$\sum_{x=16}^\infty \left(\frac{e^{-15} 15^x}{x!}\right)$$

is $$\Pr(X \ge 16)$$, the other sum is $$\Pr(X \le 15)$$.

If you have access to a cumulative Poisson probability table, or to a program that will calculate these, you can save a lot of time.