Gaussian Distribution Formula

Gaussian distribution is very common in a continuous probability distribution. The Gaussian distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables. Check out the Gaussian distribution formula below.

Formula of Gaussian Distribution

The probability density function formula for Gaussian distribution is given by,

\[\large f(x,\mu , \sigma )=\frac{1}{\sigma \sqrt{2\pi}}\; e^{\frac{-(x- \mu)^{2}}{2\sigma ^{2}}}\]

Where,

\(\begin{array}{l}x\end{array} \)
is the variable

\(\begin{array}{l}\mu\end{array} \)
is the mean

\(\begin{array}{l}\sigma\end{array} \)
is the standard deviation

Solved Examples

Question 1: Calculate the probability density function of Gaussian distribution using the following data. x = 2,

\(\begin{array}{l}\mu\end{array} \)
= 5 and
\(\begin{array}{l}\sigma\end{array} \)
= 3

Solution:

From the question it is given that, x = 2,

\(\begin{array}{l}\mu\end{array} \)
= 5 and
\(\begin{array}{l}\sigma\end{array} \)
= 3

Probability density function formula of Gaussian distribution is,

f(x,

\(\begin{array}{l}\mu\end{array} \)
,
\(\begin{array}{l}\sigma\end{array} \)
) =
\(\begin{array}{l}\frac{1}{\sigma \sqrt{2\pi }}\end{array} \)
\(\begin{array}{l}\;\end{array} \)
\(\begin{array}{l}e^{\frac{-(x-\mu )^{2}}{2\sigma ^{2}}}\end{array} \)

f(2, 5, 3 ) =

\(\begin{array}{l}\frac{1}{\sigma \sqrt{2\pi }}\end{array} \)
\(\begin{array}{l}e^{\frac{-(2-5 )^{2}}{18}}\end{array} \)

=[1/(3 ×2.51) ] × 0.6065

= 0.1328 ×0.6065

= 0.0805

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