How to find the average in the given labels?

Question

I generated 11 random variables follow the Poisson distribution. I used lambda equal to 5.

The data that I got is following :

0.00673794699908547
0.0336897349954273
0.0842243374885683
0.140373895814281
0.175467369767851
0.175467369767851
0.146222808139876
0.104444862957054
0.0652780393481588
0.0362655774156438
0.0181327887078219

Now, I got the graph as a bell curve, It looks good.

I would like to find the average value of it. I know that the average should be equal to lambda (5), however, I don't know how to find it from calculation.

I tried to sum all the data and divide by the number of data. I got 0.0897. How can I know that this value falls in the bin 5?

Answer 1

Wikipedia explains:

In the case of a discrete probability distribution of a random variable $X$, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value $x$ of $X$ and its probability $P(x)$, and then adding all these products together, giving $\sum x P(x)$.

The numbers you have appear to be the probabilities $P(x=0)$, ..., $P(x=10)$. Multiplying them by $0,1,\dots,10$ respectively and summing, I get $4.8408597 $ which is not far from $5$. The reason why this number is less than $5$ is that the probabilities you have listed do not exhaust all possible values of $x$ (which can take any nonnegative integer values). The infinite sum of $xP(x)$ over all nonnegative integers $x$ would be exactly equal to $5$.