Log-normal distribution calculator
Log-normal Distribution Calculator is used to find mean, variance and probabilities of various type of events for Log-normal distribution with parameter $\mu$ and $\sigma^2$.
Log-Normal Probability Calculator | |
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First Parameter ($\mu$) | |
Second parameter ($\sigma$) | |
P(X< A) | |
P(X > B) | |
P(A< X < B) | and |
Outside A and B | and |
Results | |
Mean : | |
Variance : | |
Required Probability : |
How to calculate probabilities of Log-normal Distribution?
Step 1 - Enter the first parameter $\mu$
Step 2 - Enter the second parameter $\sigma$
Step 3 - Select the probability type
Step 4 - Click on "Calculate" button to get Log-normal distribution probabilities
Step 5 - Gives the output of mean and variance of log-normal distribution
Step 6 - Gives the output of required probability for log-normal distribution
Definition of Log-normal Distribution
The continuous random variable $X$
has a Log Normal Distribution if the random variable $Y=\ln (X)$
has a normal distribution with mean $\mu$
and standard deviation $\sigma$
. The probability density function of $X$
is
$$ \begin{align*} f(x;\mu,\sigma) &= \begin{cases} \frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{1}{2\sigma^2}(\ln x -\mu)^2}, & x\geq 0; \\ 0, & x < 0. \end{cases} \end{align*} $$
$\mu$
is location parameter$\sigma$
is scale parameter
Normal distribution is not suitable when the data are highly skewed or data contains outliers. In such a situation, log-normal distribution is often a good choice.
The log-normal distribution is derived from a normal distribution as follows:
If $Y\sim N(\mu,\sigma^2)$ then $X= e^Y$ follows a log-normal distribution with parameter $\mu$ and $\sigma^2$.
Mean of Log-normal distribution
The mean of Log-normal distribution is $E(X) = e^{\mu+\sigma^2/2}$
.
Variance of Log-normal distribution
The variance of Log-normal distribution is $V(X) =e^{2\mu+\sigma^2}\big(e^{\sigma^2}-1\big)$
.
Log-normal Distribution Example
The life-time (in days) of certain electrionic component that operates in a high-temperature environment is log-normally distributed with $\mu=1.2$ and $\sigma=0.5$.
a. Find mean and variance of lifetime of electronic component.
b. Find the probability that the component works till 4 days.
c. Find the probability that the component works more than 5 days.
d. Find the probability that the component works between 3 and 5 days.
Solution
Let $X$ denote the life-time (in days) of certain electronic components that operates in a high-temperature environment. Given that $X\sim LN(1.2, 0.5^2)$. That is $\mu = 1.2$ and $\sigma = 0.5$.
Then $\ln(X)\sim N(1.2,0.25)$ distribution.
a. The mean of Log-normal distribution is
$$ \begin{aligned} E(X) &= e^{\mu+\sigma^2/2}\\ &= e^{1.2 + 0.5^2/2}\\ &= e^{1.325}\\ &= 3.7622 \end{aligned} $$
and the variance of log-normal distribution is
$$ \begin{aligned} V(X) &= e^{2\mu+\sigma^2}\big(e^{\sigma^2}-1\big)\\ &= e^{2*1.2 + 0.5^2}\big(e^{0.5^2}-1\big)\\ &= e^{2.65}\big(e^{0.25}-1\big)\\ &= 14.154\big(0.284\big)\\ &= 4.0197 \end{aligned} $$
b. The probability that the component works till 4 days is $P(X<4)$.
The $Z$ score that corresponds to $4$ is
$$ \begin{aligned} z&=\dfrac{\ln(X)-\mu}{\sigma}\\ &=\dfrac{\ln(4)-1.2}{0.5}\\ &\approx0.37 \end{aligned} $$
Thus the probability that the component works till 4 days is
$$ \begin{aligned} P(X < 4) &=P(\ln(X) < \ln(4))\\ &=P(Z < 0.37)\\ &=0.6443 \end{aligned} $$
c. The probability that the component works more than 5 days is $P(X>5)$.
The $Z$ score that corresponds to $5$ is
$$ \begin{aligned} z&=\dfrac{\ln(X)-\mu}{\sigma}\\ &=\dfrac{\ln(5)-1.2}{0.5}\\ &\approx0.82 \end{aligned} $$
The probability that the component works more than 5 days is
$$ \begin{aligned} P(X > 5) &=1-P(X < 5)\\ &= 1-P(\ln X < \ln (5))\\ &= 1-P(Z < 0.82)\\ &=1-0.7939\\ &=0.2061 \end{aligned} $$
d. The probability that the component works between 3 and 5 days is $P(3 < X < 5)$.
The Z score that corresponds to $3$ and $5$ are respectively
$$ \begin{aligned} z_1&=\dfrac{\ln(X)-\mu}{\sigma}\\ &=\dfrac{\ln(3)-1.2}{0.5}\\ &\approx-0.2 \end{aligned} $$
and
$$ \begin{aligned} z_2&=\dfrac{\ln(X)-\mu}{\sigma}\\ &=\dfrac{\ln(5)-1.2}{0.5}\\ &\approx0.82 \end{aligned} $$
The probability that the component works between 3 and 5 days is
$$ \begin{aligned} P(3 \leq X\leq 5) &=P(\ln (3) \leq \ln X\leq \ln(5))\\ &=P(-0.2\leq Z\leq 0.82)\\ &= P(Z < 0.82) -P( Z < -0.2)\\ &=0.7939-0.4207\\ &= 0.3732 \end{aligned} $$
Conclusion
In this tutorial, you learned about how to calculate probabilities of Log-normal distribution. You also learned about how to solve numerical problems based on Log-normal distribution.
To read more about the step by step tutorial on Log-normal distribution refer the link Log-normal Distribution. This tutorial will help you to understand Log-normal distribution and you will learn how to derive mean, variance, moments Log-normal distribution and other properties of Log-normal distribution.
To learn more about other probability distributions, please refer to the following tutorial:
Let me know in the comments if you have any questions on Log-normal Distribution Examples and your thought on this article.