Sum of random variables?

Question

If k random variables having the same probability distribution function f(x), then what is the probability distribution function of sum of these k random variables? Is it still f(x)?

Answer 1

Did you ask this in two places?  Thought I'd answered this one.

Answer depends on your level.  Choose one.

1)

Start by taking just two variables.

Start by imagining the distribution, pdf on an axis.

Take the left most value it can take, or another so we can start.

Now with that value, we want to add another for the second variable.

So the distribution for the second ADDED to the first choice is the distribution of the second one shifted along the axis so the zero of the second distribution is on the choice you made for the first one.

For example, if you choose standard normal (mean 0) then having chosen the first value, the distribution for the second one is the normal centered on the first value.

To get the pdf for the sum of two, we have to repeat this for every value of the first choice & add the pdfs.  Note that the pdf of the first variable gives us teh weighting for each first value, so we need to do it more for the values that are more likely.

So the distribution of the second is sort of smoothed out over the possible starting values for the first.

Now for each value of the first & second, do the same for the third .. k.

Clearly the distribution has to end up wider.  The distribution of the second starts, at least some of the time, on the extreme left & extreme right values of the first & so on.

The central limit theorem will ensure (under most practical cases) that the distribution tends to normal as k gets big.

2)

The description above tries to describe the convolution of the pdfs.  To find the distribution of the sum of 2 variables, convolve the pdfs.  For k variable, convolve all the pdfs.  Again the central limit theorem says the pdf will tend to normal.

3)

The Fourier transform of the pdf is called the characteristic function.

Convolution of two pdfs is equivalent to multiplying the characteristic functions.  The pdf of the sum of k variables can be calculated by the inverse Fourier transform of the product of the characteristic functions of the individual pdfs.  In the case of like pdfs, the characteristic function raised to the kth power.

Hope one of these makes sense to you.

Answer 2

Mean of sum = sum of means = k*mean of f(x)

Variance of sum = sum of variances = k*variance of f(x)

Until you give the nature of f(x) I can't take this further.

If f(x) is a Normal variable with mean m and standard deviation s, then sum of k of these is Normal with mean km and standard deviation s*sqrt(k).

Edit.

The sum of Normal distributions is itself Normal. This does not work with many other types of distribution. I'm not sure whether there are any others for which it does work.

The Central Limit Theorem says that the sum of a large number of most distributions is approximately Normal.