# Normal and exponential-normal (?) distribution

• mahtabhossain
In summary: Why? Use that ##P(Z:=X/Y < w)=P(X< yw) ; y \in Y## so that ##\int_{- \infty}^{\infty}ydy\int _{-\infty}^{yw} xdx ##defines a distribution for ##Z ## under some reasonable conditions like ## y \neq 0 ## and others. Instead of ##yw## as an integration limit, you can use any function of either, including log. Please double-check, I am on my phone here.
mahtabhossain
Dear Users,

For normally distributed random variables x and y's p.d.f.:
$$\frac{1} {\sqrt{2\pi \sigma_x^2}}\exp\left\{- \frac{(x - \mu_x)^2}{2 \sigma_x^2}\right\}$$
and
$$\frac{1} {\sqrt{2\pi \sigma_y^2}}\exp\left\{- \frac{(y - \mu_y)^2}{2 \sigma_y^2}\right\}$$

What will be the p.d.f. of ln(x) - ln(y)? Is there any method to find it?

I think the followings are the p.d.f.s of u=ln(x) and v=ln(y) given x and y are normally distributed:
$$\frac{1} {\sqrt{2\pi \sigma_x^2}}\exp\left\{u - \frac{(e^u - \mu_x)^2}{2 \sigma_x^2}\right\}$$
and
$$\frac{1} {\sqrt{2\pi \sigma_y^2}}\exp\left\{v - \frac{(e^v - \mu_y)^2}{2 \sigma_y^2}\right\}$$

I am trying to find the p.d.f. of ln(x)-ln(y). Any suggestions?

What should the difference or even ##\dfrac{X}{Y}## be? These isn't a random variable anymore.

There is a formula for the general distribution of a function of a random variable which is itself a random variable.

But how we can achieve finite distribution integrals if one variable is in the denominator? And ##ln(X)-ln(Y)## can literally take any values.

mahtabhossain said:
I think the followings are the p.d.f.s of u=ln(x) and v=ln(y) given x and y are normally distributed:

If x and y are normally distributed, they can take on negative values with non-zero probability. The expressions ln(x) and ln(y) are not defined for negative values.

WWGD
Like Stephen said, ln(x) will not be defined at the left end. Maybe you're considering ln (|x|)-ln(|y|)? Or maybe your mean is large-enough that those values are far at the tail end to not worry about them?

Or you can just add the assumption that ##X## and ##Y## are both positive.

fresh_42 said:
What should the difference or even ##\dfrac{X}{Y}## be? These isn't a random variable anymore.

Assuming ##Y\neq0## everywhere ##X/Y## is still a random variable.

Math_QED said:
Assuming ##Y\neq0## everywhere ##X/Y## is still a random variable.
It is still the wrong direction and requires additional assumptions. Instead of asking what ##\log \dfrac{X}{Y}##means, one should start whether a random variable ##Z## with properties ##\ldots## can be described by a pdf ##\ldots##

fresh_42 said:
It is still the wrong direction and requires additional assumptions. Instead of asking what ##\log \dfrac{X}{Y}##means, one should start whether a random variable ##Z## with properties ##\ldots## can be described by a pdf ##\ldots##
Why? Use that ## P(Z:=X/Y < w )=P(X< yw) ; y \in Y## so that ##\int_{- \infty}^{\infty}ydy\int _{-\infty}^{yw} xdx ##defines a distribution for ##Z ## under some reasonable conditions like ## y \neq 0 ## and others. Instead of ##yw## as an integration limit, you can use any function of either, including log. Please double-check, I am on my phone here.

Last edited:

## 1. What is a normal distribution?

A normal distribution, also known as a Gaussian distribution, is a type of probability distribution that is commonly used to describe continuous variables. It is symmetrical in shape and follows a bell-shaped curve, with the majority of data falling around the mean value.

## 2. How is a normal distribution different from other distributions?

A normal distribution is different from other distributions in that it is characterized by its mean and standard deviation, rather than specific parameters such as shape or skewness. It is also a continuous distribution, meaning that it can take on any value within a range, unlike discrete distributions which can only take on specific values.

## 3. What is the difference between a normal and exponential-normal distribution?

A normal distribution is a symmetric bell-shaped curve, while an exponential-normal distribution is a skewed distribution that combines elements of both a normal and exponential distribution. This means that it has a long tail on one side and a shorter tail on the other, and is commonly used to model data with large outliers.

## 4. How is a normal distribution used in scientific research?

A normal distribution is used in scientific research to analyze and interpret data, as it is a common assumption that many natural phenomena follow a normal distribution. It is also used in statistical tests and hypothesis testing to determine the likelihood of a certain outcome occurring.

## 5. Can a normal distribution be transformed into an exponential-normal distribution?

Yes, it is possible to transform a normal distribution into an exponential-normal distribution by applying a logarithmic transformation to the data. This can be useful when dealing with data that is skewed or has a large number of outliers, as it can help to make the data more normally distributed and easier to analyze.

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