Log Normal Distribution

Now, we will see another type of distribution which is a log normal distribution.

In this you will see even this is derived from a normal distribution.

The only new thing that is there in it, is log base.

If I say that my natural log which we call logarithm, which means ln of x.

This means that any log normal distribution is such a continuous distribution curve in which its natural log is normally distributed.

The log normal distribution’s PDF formula is as follows, which is defined in second power.

Here instead of x we have used ln of x So, mew and sigma are the parameters.

Mew is my scale parameter and sigma is my shape parameter.

Now, we will see what the curve of normal distribution and log normal distribution looks like.

Normal distribution curve, we know that it is symmetrical, which means the left section is equal to the right section, but in log normal distribution what is the difference is that it is a right skewed curve, which means since I have got all the values which are positive in log normal distribution, then this curve’s tail is extended on the right side and in this particular case all the values are mostly on the left.

So, where all is our log normal distribution used? This is a widely used curve in statistics.

It is used majorly in finance, if you want an asset return or if you want exchange rates or if we take the normal examples, if we take an example of people's income, even that forms a log normal distribution.

Why? Because there are very few people like Elon Musk and Bill Gates who have a lot of money.

So, that becomes my positive skewness case.

Always remember that since the log never takes negative value.

That's why in log normal distribution, all my values are positive.

Its mean is defined by our exponential of mew plus sigma square upon two.

Now, a very important property of log normal distribution is that if I draw a curve of random variables and PDF.

In that, I can keep the mew’s value the same and keep on changing my standard deviation, the curves that I will get will be of different types.

The way we have seen the curves in normal distribution, in the same way even these will get flattened and widened and if my standard deviation is the lowest then my curve will be wide and as and when my standard deviation will keep on increasing.

In the same way, my curve’s tail will also increase.

And my curve will become a more positive skew.

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