Key Properties and Applications of Lognormal Distribution
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.
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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.
FAQs on Lognormal Distribution Explained for Students
1. What is a lognormal distribution in simple terms?
A lognormal distribution is a continuous probability distribution for a random variable whose logarithm is normally distributed. In simpler terms, if you take a set of data that follows a lognormal pattern and calculate the natural logarithm of each data point, the resulting new set of data will form a symmetrical, bell-shaped normal distribution. This distribution is only defined for positive real numbers.
2. What is the main difference between a lognormal and a normal distribution?
The primary difference lies in their shape and the range of values they can model. A normal distribution is symmetrical (bell-shaped) and can take any real value, including negative ones. In contrast, a lognormal distribution is positively skewed (skewed to the right) with a long tail and can only model variables that are greater than zero. This makes it ideal for quantities that cannot be negative, like stock prices or income.
3. What are the key parameters that define a lognormal distribution?
A lognormal distribution is defined by two key parameters, which are actually the parameters of the associated normal distribution of its logarithm:
μ (mu): This is the mean of the variable's natural logarithm (not the mean of the variable itself).
σ (sigma): This is the standard deviation of the variable's natural logarithm.
These two parameters determine the location and shape of the lognormal curve.
4. Can you provide a real-world example of lognormal distribution?
A classic real-world example is the distribution of household income in an economy. Most households earn a low to moderate income, creating a large peak on the left side of the graph. However, a small number of households earn extremely high incomes, which creates a long tail extending to the right. Other common examples include the price of stocks, the population of cities, and the length of comments on social media.
5. What is the formula for the Probability Density Function (PDF) of a lognormal distribution?
The formula for the Probability Density Function (PDF) of a lognormal distribution for a variable x is given by:
f(x; μ, σ) = [1 / (x * σ * √(2π))] * e-((ln(x) - μ)² / (2σ²))
This formula is valid for x > 0. In this equation, 'μ' is the mean and 'σ' is the standard deviation of the variable's natural logarithm, and 'e' is Euler's number (approximately 2.718).
6. Why is the lognormal distribution always skewed to the right?
The lognormal distribution is always right-skewed because of the exponential relationship between the normally distributed logarithm and the lognormally distributed variable itself. The exponential function amplifies the differences between larger numbers more than smaller numbers. Therefore, when the symmetrical values of a normal distribution are transformed, the values on the higher end are stretched out much more than the values on the lower end. This creates a distribution with a lower bound at zero and an infinitely long tail to the right.
7. How are the mean and variance of a lognormal distribution calculated?
The mean and variance of a lognormal distribution are not simply its parameters μ and σ. Instead, they are calculated from these parameters as follows:
Mean (Expected Value) = e(μ + σ²/2)
Variance = (e(σ²) - 1) * e(2μ + σ²)
This shows that both the mean and variance depend on both μ and σ of the underlying logarithmic data.
8. In what situations is using a lognormal distribution more appropriate than a normal distribution?
A lognormal distribution is more appropriate in situations where:
The variable cannot be negative (e.g., price, height, weight).
The data is positively skewed, with most values being clustered at lower levels but with a few very high values.
The underlying mechanism involves multiplicative growth or effects, rather than additive ones. For example, investment returns compound multiplicatively, making stock prices well-suited for a lognormal model.
Using a normal distribution for such data could lead to illogical predictions, such as a negative stock price.
9. If a variable X follows a lognormal distribution, what does this imply about its logarithm, ln(X)?
This is the core definition of the distribution. If a variable X follows a lognormal distribution, it directly implies that its natural logarithm, ln(X), follows a normal distribution. The parameters used to describe the lognormal distribution of X (μ and σ) are, in fact, the mean and standard deviation of the normal distribution followed by ln(X). This transformation is the key to analysing and understanding lognormally distributed data.
