Lessons

- 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

# Discrete Uniform Random Variable

A discrete uniform random variable is a discrete random variable for which the probability of each outcome is the same.

**Example**

The roll of a die is a discrete uniform random variable and has a discrete uniform probability distribution.

The random variable X can take the values X = {1, 2, 3, 4, 5, 6}

Each outcome has a probability of 1/6.

The probability distribution and cumulative distribution functions are shown below:

xi | Probability distribution, P(xi) | Cumulative Distribution, F(xi) |

1 | 1/6 | 1/6 = 0.1667 |

2 | 1/6 | 2/6 = 0.333 |

3 | 1/6 | 3/6 = 0.5 |

4 | 1/6 | 4/6 = 0.667 |

5 | 1/6 | 5/6 = 0.833 |

6 | 1/6 | 6/6 = 1 |

Let’s observe a few values from the above table.

P(3) = 1/6 or 0.1667

F(3) = 0.5

P(2<=X<=5) = 4*1/6 = 0.667

We can generalize this as follows:

P(x) = 1/6 or 0.1667

Cumulative distribution function for any outcome i is F(xi) = i*P(x)

Probability function for a range with k outcomes = k*P(x)