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Normal Distribution Formula

In probability theory, the normal or Gaussian distribution is a very common continuous probability distribution.

A normal distribution is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, lengths of objects produced by machines, etc.

The spread of a normal distribution is controlled by the standard deviation, $\sigma$.  The smaller the standard deviation the more concentrated the data.

The formula for normal probability distribution is given by:

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

$\mu$ = Mean of the data
$\sigma$ = Standard Distribution of the data.
When mean ($\mu$) = 0 and standard deviation($\sigma$) = 1, then that distribution is said to be normal distribution.
x = Normal random variable.

Solved Examples

Question: An average light bulb lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that the light bulb will last at most 365 days?


A mean score of 300 days and a standard deviation of 50 days, we want to find the cumulative probability that bulb life is less than or equal to 365 days. Thus, we know the following:

  • The value of the normal random variable is 365 days.
  • The mean is equal to 300 days.
  • The standard deviation is equal to 50 days.

The answer is: P( x < 365) = 0.90. Hence, there is a 90% chance that a light bulb will burn out within 365 days.

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