Lognormal Distribution Formula Properties and How to Solve Problems
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 Statistics Students
1. What is a lognormal distribution?
A lognormal distribution is a probability distribution of a random variable whose natural logarithm is normally distributed. In other words, if X is lognormally distributed, then ln(X) follows a normal distribution.
- It is defined only for X > 0.
- It is positively skewed (right-skewed).
- It is commonly used to model growth processes, incomes, stock prices, and biological measurements.
2. What is the formula for the lognormal distribution?
The probability density function (PDF) of a lognormal distribution is given by f(x) = (1 / (xσ√(2π))) · exp(−(ln x − μ)² / (2σ²)), for x > 0.
- μ = mean of ln(X)
- σ = standard deviation of ln(X)
- x > 0
3. What are the mean and variance of a lognormal distribution?
The mean of a lognormal distribution is E(X) = e^(μ + σ²/2) and the variance is Var(X) = (e^(σ²) − 1)e^(2μ + σ²).
- μ and σ are the mean and standard deviation of ln(X).
- The mean is always greater than the median due to right skewness.
4. What is the difference between normal and lognormal distribution?
The key difference is that a normal distribution is symmetric over all real numbers, while a lognormal distribution is right-skewed and defined only for positive values.
- Normal: X ~ N(μ, σ²), values can be negative or positive.
- Lognormal: ln(X) ~ N(μ, σ²), so X > 0.
- Normal is symmetric; lognormal is positively skewed.
5. How do you know if a variable follows a lognormal distribution?
A variable follows a lognormal distribution if its logarithm is normally distributed. To check this:
- Take Y = ln(X).
- Test whether Y follows a normal distribution using a histogram or normal probability plot.
- Check for right-skewed shape in the original data.
6. What is the median and mode of a lognormal distribution?
The median of a lognormal distribution is e^μ and the mode is e^(μ − σ²).
- μ is the mean of ln(X).
- The median is less than the mean.
- The mode is the peak (most frequent value) of the distribution.
7. Can you give an example of a lognormal distribution?
An example of a lognormal distribution is stock prices in financial markets.
- If daily returns are normally distributed,
- Then stock prices (which grow multiplicatively) follow a lognormal distribution.
8. How do you calculate probabilities in a lognormal distribution?
To calculate probabilities for a lognormal distribution, convert the variable using the natural logarithm and use the normal distribution table. Steps:
- Step 1: Let Y = ln(X).
- Step 2: Standardize using Z = (ln(x) − μ)/σ.
- Step 3: Use the standard normal table to find the probability.
9. Why is the lognormal distribution always positively skewed?
A lognormal distribution is always positively skewed because exponentiating a normal variable stretches values more on the right side.
- If Y is normal, X = e^Y.
- Large positive Y values produce very large X values.
- X cannot be negative.
10. Where is the lognormal distribution used in real life?
The lognormal distribution is used to model real-life quantities that grow multiplicatively and cannot be negative. Common applications include:
- Stock prices and financial modeling
- Income and wealth distribution
- Size of firms or cities
- Biological growth and survival times
