- 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
Cumulative Distribution Function
We can also construct a cumulative distribution function for a random variable. A cumulative distribution function gives the probability that the random variable X is less than or equal to x, for every value x. In case of discrete random variables, the cumulative distribution function is the sum of the probabilities of all outcomes unto and including the specific outcome x.
The cumulative distribution function is expressed as .
We will build upon our earlier probability distribution example.
| xi | P(xi) | F(xi) |
| 1 | 0.2 | 0.2 |
| 2 | 0.3 | 0.5 |
| 3 | 0.4 | 0.9 |
| 4 | 0.1 | 1 |
Probability that X =1 is 0.2
Probability that X = 1 or 2 = 0.2 + 0.3 = 0.5
Probability that X = 1 or 2 or 3 = 0.2 + 0.3 +0.4 = 0.9
Probability that X = 1 or 2 or 3 or 4= 0.2 + 0.3 +0.4 +0.1= 1.0
The histogram for cumulative distribution will look as follows:
The above cumulative distribution was for a discrete random variable. Even a continuous random variable will have a cumulative distribution function.
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