[Maths Class Notes] on Lognormal Distribution Pdf for Exam

The term lognormal distribution in probability theory is defined as a continuous probability distribution of random variable whose logarithm values are normally distributed. The lognormal distribution is also known as a logarithmic normal distribution. For example,  If a random variable X is considered as the log-normally distributed then Y = In(X) will have a normal distribution. Similarly, if it is determined as a normal distribution then the exponential function of Y, X= esp (Y), has a log-normal distribution. A random variable that is a log-normally distributed considers only positive real values. It is the easiest and useful model for measurements in engineering science, as well as in medical and economics.

Lognormal distribution exhibits phenomena whose relative growth is independent of its size, which is valid in most natural phenomena comprising the size of the tissue, blood pressure, income distribution, and even in the length of the game of chess.

Log Normal Distribution Definition

Let Z be a standard normal variable, which implies that the probability distribution of Z is normally centered at 0 and with variance 1. Then a log -normal distribution is defined as the probability distribution of random variables.

X = eμ + σZ

Where μ and σ are represented as the standard deviation of the logarithm of X

The term log normal is derived from the result of taking logarithm from both sides of the equation.

Log X = μ + σZ

As Z is a normal distribution, μ + σZ is also considered as a also a normal distribution (this transformation does not affect normality rather just scales the distribution). It implies that the logarithm of X is normally distributed.

How to Find Log – Normal Distribution?

Lognormal distribution satisfies the following equation:

fₓ(x) = 1/ σx2e ⁻ (In x –μ)²/ 2σ²

which is a result of change of variable theorem and small number of calculus. As, this form is more complicated to use with hand, so it is more beneficial to use the general properties of distribution i.e.(eg. mean and tendencies). For this reason, it is beneficial to examine the results when mean =  0, and standard deviation = 1.

The above distribution is skewed to the left and the mode is approximately .35 ( in fact, it is 1/e). This along with the general shape of the curve, is quite sufficient information to form a reasonably accurate approximation of the curve.

Lognormal Distribution Parameters

The following lognormal distribution parameters are responsible for the shape of a log-normal distribution:

Shape Parameter – The standard deviation of log normal distribution affects the general shape of the distribution. The shape parameter does not change the location or height of the graph, rather it just affects the shape of the graph.

Scale Parameter – It is a median that tends to shrink or stretch the graph.

Location  Parameter – It informs us where the graph is placed on the x-axis.

Mean of Lognormal Distribution

The mean of log – normal distribution is given as 

m= eμ +σ²/2

which also implies that μ can be calculate from m:

μ = Inm – ½ σ²

Median of Lognormal Distribution

The median of the log – normal distribution is  Med [X] = eμ

which is obtained by setting the cumulative frequency equals to 0.5 and solving the resulting equation.

Mode of Lognormal Distribution

The mode of the log-normal distribution is  stated as:

Mode [X] = eμ – σ²

which is obtained by setting the probability distribution function equal to 0, as the mode represents the global maximum of the distribution.

Variance of Lognormal Distribution

The variance of the log – normal distribution is Var [X] = (eσ² – 1)e2μ + σ²

which can also be written as  (eσ² – 1) where m represents the mean of the distribution above.