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Log-normal distribution for the prices 🤑
Recall that in our last case study, we modeled the first digit of the sample of supermarket prices with the Benford’s law. This time, let us model the prices as realizations of a continuous random variable \(X\) following a continuous distribution law.
What do you think the distribution law should be like? Can it be a normal distribution? An exponential distribution? Or do we need some other distribution to describe the associated random variable appropriately?
Let’s look at the histogram of the \(50\) prices:
Obviously, log-normal distribution would be the best choice for our model. But what is this “log-normal distribution”?
Log-normal distribution
” a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable \(X\) is log-normally distributed, then \(Y = \ln(X)\) has a normal distribution.”
So,
- if a random variable \(X\) is log-normally distributed, then \(\ln X\) is normally distributed.
- Since the function \(\ln(\cdot)\) admits only positive values, the log-normal distribution is a suitable model for positive-valued real random variables.
The density of the log-normal distribution for \(x \in(0;\infty)\) is:
\[f(x)=\frac1{x\sigma\sqrt{2\pi}}\cdot e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}\]
with the resulting \(\mathbb E(X)=e^{\mu+\frac{\sigma^2}2}\) and \(\text{Var}(X)=(e^{\sigma^2}-1)\cdot e^{2\mu +\sigma^2}.\)
Below, you can find the densities of log-normal distributions with different parameter values.
😎 So, if our prices follow a log-normal distribution, we can apply the natural logarithm transformation to the prices and model them subsequently using a normal distribution! 😎
Normal distribution for the log-prices 🔬
Now recall our real sample of \(50\) prices for you here (https://www.edeka24.de)
Log-normal distribution
\[prices\sim \log N(\mu,\sigma^2)\rightarrow \ln (prices)\sim N(m,s^2)\]
A sample of real prices (no intended advertisement). Click here to unfold/fold the data 📲
So, if our prices follow a log-normal distribution, we can apply the
natural logarithm transformation to the prices, which
are already added in R as a variable prices, and model them
subsequently using a normal distribution!
Follow the following steps:
- compute the natural logarithm of the sample prices and call the new
variable
lprices. Use the R-functionlog(prices)or=LN()in excel/calc.
If you wish, you can use the following chunk to compute the probability in the question above using R:
lprices = log(prices)Useful R-functions
m = mean(...)to compute the sample mean,s = sd(...)to compute the sample standard deviation,dnorm(x,m,s)to compute the density in points \(x\) modeled with normal distribution with mean \(m\) and standard deviation \(s\).pnorm(x,m,s)to compute the the probability \(\mathbb P(X\leq x)\) for \(X\) following a normal distribution with mean \(m\) and standard deviation \(s\).
- compute the sample mean
mand the sample standard deviationsof the log-prices:
m = mean(lprices)
s = sd(lprices)
# print out the results
print(paste("the sample mean is",round(m,4),"and the standard deviation is",round(s,4)))- plot the density of the resulting normal distribution. Use R-command
x=seq(from=-5,to=5,by=0.01)to create a sequence of values, andfx=dnorm(x,mean=m,sd=s)to evaluate the density at the values. You can useplot(x,fx,type="l")to show a plot.
x = seq(-5,5,0.01) # sequence of values
fx = dnorm(x,m,s) # density at the values
plot(x,fx, type="l") # plot of type "l"=lines- compute the following probabilities according to the log-normal model:
If you wish, you can use the following chunk to compute the
probability in the question above using R (Note that the function
pnorm(number,m,s) will compute for \(\mathbb P(X\leq ~number)\) for \(X\sim N(m,s^2)\)).
pnorm(3,m,s)
pnorm(3,m,s)- pnorm(1,m,s)
pnorm(log(3),m,s)
pnorm(log(3),m,s)- pnorm(log(1),m,s)