- 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

# What is a Probability Distribution

We know that a random variable is an uncertain quantity or a number. Its value is determined by chance. For example, the outcome of rolling a die is random. We could get any number from 1 to 6. In case of a die, the probability of getting any number is 1/6. Each outcome has the same probability. However, the probability of each outcome could be different.

A probability distribution is a graph or a table that describes the probabilities of each outcome of a random variable.

In a probability distribution, each value or outcome of the random variable is represented as x. The probability of getting x is represented as P(x). So, if X is the random variable, we are saying that the probability of random variable X being equal to x is P(X=x) or P(x). This is called the probability function.

The probability distribution of rolling a die is shown below:

xi | P(xi) |

1 | 1/6 |

2 | 1/6 |

3 | 1/6 |

4 | 1/6 |

5 | 1/6 |

6 | 1/6 |

Note that the sum of all probabilities should be equal to 1 and the probability of each outcome, P(x) is between 0 and 1.

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