Lesson 47 – Is it normal?

The wait time for the first/next arrival follows an Exponential distribution. The wait time for the ‘r’th arrival ( T_{r} = t_{1} + t_{2} + t_{3} + ... + t_{r} ) follows a Gamma distribution.

The probability density function of the Gamma distribution is derived using the convolution of individual random variables  t_{1}, t_{2}, ... t_{r} .

 f(t) = \frac{\lambda e^{-\lambda t}(\lambda t)^{r-1}}{(r-1)!}

For increasing values of r, the distribution is like this.

It tends to look like a bell. Is it normal?

Nah, it may be a Gamma thing. Let me add uniform distributions.

 f(x) = 1 \forall 0 < x < 1

For increasing values of n, the distribution of the sum of the uniform random variables is like this.

It tends to look like a bell. Is it normal?

Hmm. I think it is just a coincidence. I will check Poisson distribution for increasing values of \lambda. Afterall, it is a discrete distribution.

P(X=x) = \frac{e^{-\lambda t}(\lambda t)^{x}}{x!}; x = 0, 1, 2, ...

Tends to look like a bell. Is it normal?

Perhaps coincidence should concede to a consistent pattern. If this is a pattern, does it also show up in the Binomial distribution?

P(X=x) = {n \choose x}p^{x}(1-p)^{n-x}; x = 0, 1, 2, ... n

There it is again. It looks like a bell.

What is this? Is it normal?

The shape is limited to a bell. Is it normal?

It is the same for any variable. Is it normal?

Why is it normal?

What is the normal?

To be continued…

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