# The PDF of the exponential of a Gaussian random variable

• samuelandjw
In summary, the PDF of the exponential of a Gaussian random variable is given by an integral involving the Dirac delta function and the PDF of the original variable. This can be simplified using the properties of the Dirac delta function and the Jacobian of the transformation. The resulting PDF is almost a log-normal distribution, but a slight discrepancy arises when y is constant in the integral.
samuelandjw
What is the PDF of the exponential of a Gaussian random variable?

i.e. suppose W is a random variable drawn from a Gaussian distribution, then what is the random distribution of exp(W)?

Thank you!

The PDF for:
$$Y = \exp{\left(\alpha \, X\right)}$$
where the PDF for $X$ is:
$$\varphi_{X}(x) = \frac{1}{\sqrt{2 \, \pi} \, \sigma} \, e^{-\frac{(x - a)^{2}}{2 \, \sigma^{2}}}$$
is given by:
$$\varphi_{Y}(y) = \int_{-\infty}^{\infty}{\delta(y - \exp(\alpha \, x)) \, \varphi_{X}(x) \, dx}$$

Use the properties of the Dirac delta function to evaluate this integral exactly!

Dickfore said:
The PDF for:
$$Y = \exp{\left(\alpha \, X\right)}$$
where the PDF for $X$ is:
$$\varphi_{X}(x) = \frac{1}{\sqrt{2 \, \pi} \, \sigma} \, e^{-\frac{(x - a)^{2}}{2 \, \sigma^{2}}}$$
is given by:
$$\varphi_{Y}(y) = \int_{-\infty}^{\infty}{\delta(y - \exp(\alpha \, x)) \, \varphi_{X}(x) \, dx}$$

Use the properties of the Dirac delta function to evaluate this integral exactly!

Thank you, Dickfore.

I realize I can just use $$\varphi_{Y}(y)=\varphi_{X}(x) \frac{dx}{dy}$$
and the result is almost log-normal distribution pdf, but your method looks quite interesting.

I do have a question, in your last integral, suppose $$y$$ is constant in the integral, then the result would be $$\varphi_{Y}(y)=\varphi_{X}(x=\frac{1}{\alpha}\ln{y})$$, which is not quite the same as the log-normal. I'm not sure if I have gotten it wrong.

you had forgotten the Jacobian of the transofrmation:
$$\left|\frac{d x}{d y}\right|$$
expressed as a function of y. For what values of y do the above expressions make sense?

Dickfore said:
you had forgotten the Jacobian of the transofrmation:
$$\left|\frac{d x}{d y}\right|$$
expressed as a function of y. For what values of y do the above expressions make sense?

Thx.

## 1. What is the PDF of the exponential of a Gaussian random variable?

The PDF (Probability Density Function) of the exponential of a Gaussian random variable is a mathematical expression that describes the probability of the exponential of a Gaussian random variable taking on a specific value. It is often used in statistics and probability to model the behavior of random variables.

## 2. How is the PDF of the exponential of a Gaussian random variable calculated?

The PDF of the exponential of a Gaussian random variable can be calculated using the formula: f(x) = (1/σ) * (1/√(2π))*e^(-(x-μ)/σ), where μ is the mean of the Gaussian random variable and σ is the standard deviation.

## 3. What is the significance of the PDF of the exponential of a Gaussian random variable?

The PDF of the exponential of a Gaussian random variable is significant because it allows us to understand the distribution of values that the exponential of a Gaussian random variable can take on. This can be useful in predicting the likelihood of certain outcomes in statistical analyses.

## 4. Can the PDF of the exponential of a Gaussian random variable be used to calculate probabilities?

Yes, the PDF of the exponential of a Gaussian random variable can be used to calculate probabilities by integrating over a certain range of values. This allows us to determine the likelihood of a certain outcome occurring within that range.

## 5. How does the PDF of the exponential of a Gaussian random variable differ from the PDF of a regular Gaussian random variable?

The main difference between the two is that the exponential of a Gaussian random variable is a transformation of a regular Gaussian random variable. This means that the exponential of a Gaussian random variable has a different shape and distribution compared to a regular Gaussian random variable.

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