# Poisson Distribution

#### samot79

##### New Member
Hi,

I am using the Poission distribution to calculate probabillity for x = X.

Poisson distribution is skewed to the right when lambda is small.

Problem I have is that distribution for my x is rather skewed to the left when the expectancy.

Is there any other known distribution I can use to calculate x = X, that is skewed to the left when expectancy for x is small?

Is it is possible to use the binomial distribution where I use p as "skewness parameter" and n be 'expected value'/prob?

Really appreciate som help for this.

#### samot79

##### New Member

My variable x is discrete and can only take whole values. Thats why I first thought Poission would be so great.

Beta distributed variable can only take values between 0-1, unless I miss something here?

How would you apply beta distribution to calculate probabillity that x >= 4 if expected value for x = 3,1 and x~beta(10, 3)?

#### rogojel

##### TS Contributor
Well, I could normalize the variable -by dividing all values by the maximum - so instead of x>4 I would look at the probability of x>0.5 if the maximum value is 8, say.

regards

#### Dason

##### Ambassador to the humans
Can you post a histogram of your data?

#### samot79

##### New Member
Hi,

@rogojel - this still feels wrong since beta distributed variable is continuously and my variable is not. There is also no upper limit on my variable.

What I am trying to do is to fit a distribution for a basketplayers points.

Expected points for my player is easy;

E(Points) = E(x) * 1 + E(y) * 2 + E(z) * 3

I guess variance of points will be;

Var(Points) = E(x) * 1^2 + E(y) * 2^2 + E(z) * 3^2

Points will surely not follow a poisson distribution since Var(points) > E(Points). One way around this would be to do simulations of x, y and z and then simply count number of times points > some value. I would however prefer to calculate that chance analytic.

Edit:
Histogram of real data seems to be a bit skewed to left, so x, y and z are probably not following a poisson distribution at all.

#### Dragan

##### Super Moderator
In view of the above, I would point out to you that there are several variations of the basic Poisson discrete distribution. From what you are describing it appears to be a "Poisson-Stopped-Sum Distribution."

That said, there are several variations of the (Poisson) distribution noted above. The general theory gives special forms such as: Hermite, Neyman Type A, Ploya-Aeppli, and the Lagrangian-Poisson distributions.

For further details see: Johnson, N., Kotz, S, & Kemp, A. W. (1993). Univariate Discrete Distributions (2-nd edition), Wiley & Sons, New York.

#### samot79

##### New Member
Sorry for my late reply.

Thanks for your reply Dragan, I have not heard of "Poisson-Stopped-Sum" and will definitly look it up.

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