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LogNormalDistribution

LogNormalDistribution
represents a lognormal distribution derived from a normal distribution with mean and standard deviation .
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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Generate a set of random numbers that are lognormally distributed:
Compare the histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare the density histogram of the sample with the PDF of the estimated distribution:
Skewness grows exponentially with standard deviation :
Limiting values:
Kurtosis grows exponentially with standard deviation :
Limiting values:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Hazard function:
Quantile function:
GammaDistribution data can be approximated by a lognormal distribution:
Comparing log-likelihoods with estimation by gamma distribution:
Lognormal distribution was traditionally used to analyze the fractional stock price changes from the previous closing price. Find the estimated distribution for the daily fractional price changes of the S&P 500 index from January 1, 2000 to January 1, 2009:
To fit lognormal distribution, all data must be positive:
Compare the histogram of the data with the PDF of the estimated distribution:
Find the probability of the fractional price change being greater than 0.5%:
Find the mean fractional price change:
Simulate fractional price changes for 30 days:
Show that using LogisticDistribution provides better fit than using lognormal distribution:
Lognormal distribution can be used to model stock prices:
Fit the distribution to the data:
Compare the histogram to the PDF:
Find the probability that the price is above $500:
Find the mean price:
Simulate the price for the consecutive 30 days:
Lognormal distribution can be used to approximate wind speeds:
Find the estimated distribution:
Compare the PDF to the histogram of the wind data:
Find the probability of a day with wind speed greater than 30 km/h:
Find the mean wind speed:
Simulate wind speeds for a month:
The fractional change of stock price at time (in years) is assumed lognormally distributed with parameters and :
Compute expected stock price at epoch :
Assuming an investor can invest money for a year at a continuously compounded yearly rate risk-free, the risk-neutral pricing condition requires:
Solve for parameter :
Consider an option to buy this stock a year from now, at a fixed price . The value of such an option is:
The risk-neutral price of the option is determined as the present value of the expected option value:
Assuming rate of 5%, volatility parameter of 0.087, an initial price of $200 per share of stock, and a strike price of $190 per share, the Black-Scholes option price is:
Parameter influence on the CDF for each :
Lognormal distribution is closed under scaling by a positive factor:
Power of a LogNormalDistribution follows a lognormal distribution:
In particular, a reciprocal of a lognormal distribution follows a lognormal distribution:
The product of two independent lognormally distributed variates follows lognormal distribution:
Quotient of two independent lognormally distributed variates follows lognormal distribution:
Geometric mean of independent identically lognormally distributed variates follows lognormal distribution:
Relationships to other distributions:
NormalDistribution is exponentially related to LogNormalDistribution:
Reverse transformation:
Lognormal distribution is a special case of SL JohnsonDistribution:
SuzukiDistribution can be obtained from lognormal distribution and RayleighDistribution:
LogNormalDistribution is not defined when is not a real number:
LogNormalDistribution is not defined when is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
LogNormalDistribution is not uniquely determined by its sequence of moments:
Compute the sequence of moments:
Compare it to the sequence of moments of the LogNormalDistribution:
Plot distribution densities:
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