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:
\[\large P(x)=\frac{1}{\sqrt{2\pi \sigma^{2}}}\:e^{\frac{(x-\mu)^{2}}{2\sigma ^{2}}}\]
Where,
$\mu$ = 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?
Solution:
Given:
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.
