- What is a Probability Distribution
- Discrete Vs. Continuous Random Variable
- Cumulative Distribution Function
- Discrete Uniform Random Variable
- Bernoulli and Binomial Distribution
- Stock Price Movement Using a Binomial Tree
- Tracking Error and Tracking Risk
- Continuous Uniform Distribution
- Normal Distribution
- Univariate Vs. Multivariate Distribution
- Confidence Intervals for a Normal Distribution
- Standard Normal Distribution
- Calculating Probabilities Using Standard Normal Distribution
- Shortfall Risk
- Safety-first Ratio
- Lognormal Distribution and Stock Prices
- Discretely Compounded Rate of Return
- Continuously Compounded Rate of Return
- Option Pricing Using Monte Carlo Simulation
- Historical Simulation Vs Monte Carlo Simulation

# Normal Distribution

The normal distribution is the well-known bell-shaped curve depicted below. The bell-shaped curve comes from a statistical tendency for outcomes to cluster symmetrically around the mean (or average).

Deviations from the mean are described in terms of standard deviations. In all normal distributions, 68% of outcomes will fall within 1 standard deviation to either side of the mean.

Let’s illustrate the concept of mean and standard deviation with a simple example. My New York subway commute every day is 30 minutes on average, with a standard deviation of 5 minutes.

Assuming a normal distribution for the time it takes me to get to work, this would imply that:

68% of the time, I can expect my daily commute to be between 25 minutes and 35 minutes (i.e., the mean of 30 minutes plus or minus 1 standard deviation, or 5 minutes).

16% of the time, my commute is less than 25 minutes (because the normal distribution is symmetrical around the mean, I expect this event to occur 16% of the time, or (100%-68%)/2).

16% of the time, my commute is greater than 35 minutes (again, because the normal distribution is symmetrical). Or, in other words, my 84% confidence level worst-case commute is 35 minutes (Only 16% of the time I would expect longer commute).

From this example, it makes sense that the more standard deviations we move from the mean, the lower the probability is of such an event occurring. For example, a delay of 10 minutes or more (2 standard deviations) only has a 2.5% chance of occurring, compared to a 16% probability of a delay of 5 minutes or more (1 standard deviation).

The table below relates standard deviations to lower tail probabilities (lower tail probabilities quantify the chance of an event of that magnitude or greater occurring):

Standard Deviations | Lower Tail Probability | Commuting Example |

1 | 16% | Delay of 5 minutes or more |

1.28 | 10 | Delay of 6.4 minutes or more |

1.65 | 5 | Delay of 8.25 minutes or more |

2 | 2.5 | Delay of 10 minutes or more |

2.33 | 1 | Delay of 11.65 minutes or more |

**Characteristics of Normal Distribution**

- The distribution is characterized by bell curve which has more weight in the center and tapers off on either side which means it has tails on either side. It takes only two moments i.e. the mean µ and the variance σ2 to describe this function and is therefore a parametric function.
- The mean gives more information about the location and the variance gives an idea of how dispersed the values are.
- A normal distribution has a skewness=0 and kurtosis=3. The excess kurtosis is 0.
- A linear combination of 2 or more normally distributed random variables is also normally distributed. This means if X and Y are normally distributed then X + Y is also normally distributed.

The density function for a normal distribution is given using the following formula:

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