Lognormal Distribution: Overview, Questions, Preparation

The normal distribution is a resultant of symmetric probability distribution and the probability distribution, which has a normally distributed logarithm for continuous random variables, known as a lognormal distribution. There are only positive real values in the lognormal distribution of random variables. 

The lognormal distribution is a persistent probability distribution for a random variable with a normal distribution of the logarithm. Thus, Y = ln(X) is normally distributed only when the random variable X has a normal distribution. Therefore, the exponential function X = exp(Y) has a lognormal distribution where Y is also normally distributed.

Properties of the lognormal distribution

Following are the properties of lognormal distribution:

Geometric moments - Lognormal’s geometric mean is expressed as GM[X] = e^µ= µ* and the standard deviation for lognormal distribution is given by GSD[X] =eσ= σ*

Coefficient of variation - CV[X] is the arithmetic coefficient of variation, given as SD[X]/E[X]. The same for lognormal distribution is represented as CV[X] = √e𝜎² -1

Mode - The mode of the lognormal distribution is represented as Mode[X] = e^{μ 𝜎²}

Median - The median of the lognormal distribution is represented as Med[X] = e^µ= µ*

Lorenz curve - The arithmetic A, geometric G, and harmonic H means of symmetrical Lorenz curve lognormal distribution are related, and the relation is expressed as H = G²/A

Probability density function

Let us consider a lognormally distributed positive random variable X. Therefore, the general formula to represent the probability density function is:

fx(x) = 1/ x𝜎√2π exp[-(ln x - μ)²/ 2ර²

Applications of the lognormal distribution

Following are the applications of the lognormal distribution

1. Lognormal distribution helps to analyse the time spent by active users on online activity.
2. It also helps in the income analysis of a certain population.
3. It also helps to decode the stock market fluctuations.
4. We can also find lognormal distribution in the length of a social media comment made by a user.
5. The presence of lognormal distribution also helps us to solve a Rubik’s Cube.
6. The maximum value of annual rainfall is also calculated using a lognormal distribution.
7. The lognormal distribution is also used to estimate the size of a biological specimen.

The topic lognormal distribution is from Chapter 13 - Probability. Apart from this topic, the chapter contains Bayes’ theorem, multiplication theorem on probability independent events, and total probability. Moreover, the chapter accounts for a pivotal 8 marks in the 12^th standard final examinations.

1. What is the probability distribution of getting heads in two tosses of a coin?
Solution.
The given condition states that a coin is tossed twice.
Therefore,  S = {HH, HT, TH, TT}
Y expresses the number of heads.

⇒ Y (HH) = 2

Y (HT) = 1
Y (TH) = 1
Y (TT) = 0

Therefore, Y is a function ranging {0, 1, 2}.

Thus, Y is a random variable that can give values in the form of 0, 1, or 2.

Now,

P (HH) = P (HT) = P (TH) = P (TT) = 1/4
P (Y = 0) = P (TT) = 1/4
P (Y = 1) = P (HT) + P (TH) = 1/4 + 1/4 = 1/2
P (Y = 2) = P (HH) = 1/4

Hence, the required probability distribution is,

Q: Why is the lognormal distribution used so extensively?

Q: What are the two parameters of lognormal distribution?

Q: What is the other name of lognormal distribution?

Q: Who are the personalities associated with lognormal distribution?

Q: What is the other name of the central limit theorem?