This calculator computes probabilities for the lognormal distribution using the mean (𝝁), standard deviation (𝝈), and a random variable value (𝑥). It also visualizes the distribution and highlights the cumulative probability \( P(X \leq x) \).
P(X = ):
P(X < ):
P(X ≤ ):
P(X > ):
P(X ≥ ):
Understanding the Lognormal Distribution
The Lognormal Distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. It is widely used in fields like finance, reliability analysis, and biological modeling.
In particular, the Black-Scholes Model assumes that the price of financial assets follows a lognormal distribution, which allows it to model stock prices effectively over time.
Key Components of the Lognormal Distribution
- Mean (𝝁): The mean of the underlying normal distribution.
- Standard Deviation (𝝈): The standard deviation of the underlying normal distribution.
- Random Variable (𝑥): The value for which probabilities are being calculated.
Formula for Lognormal Distribution
The probability density function (PDF) for the lognormal distribution is:
Example Calculation
Let's assume the mean is \( \mu = 0 \), the standard deviation is \( \sigma = 1 \), and we want to find the probability for \( x = 1 \).