Continuous Random Variable: Difference between revisions
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distribution, where every point has the same probability density. | distribution, where every point has the same probability density. | ||
PDF | |||
<math> | <math> | ||
f(x) = \begin{cases} | f(x) = \begin{cases} | ||
\frac{ 1 }{ b - a } & a \leq x \leq b \\ | \frac{ 1 }{ b - a } & a \leq x \leq b \\ | ||
0 & otherwise | 0 & \text{otherwise} | ||
\end{cases} | \end{cases} | ||
</math> | |||
CDF | |||
<math> | |||
f(x) = \begin{cases} | |||
0 & x < a \\ | |||
\frac{ x - a }{ b - a } & a \leq x \leq b \\ | |||
1 & x > b | |||
\end{cases} | |||
</math> | |||
Mean | |||
<math> | |||
\mu_X = \frac{ a + b }{ 2 } | |||
</math> | |||
Variance | |||
<math> | |||
\frac{ 1 }{ 12 } (b - a)^2 | |||
</math> | </math> |
Revision as of 07:55, 1 March 2024
Continuous random variables have an inifinite number of values for any given interval. While similar, the approach to analysis is very different from discrete variables
- Summation becomes integration
- Probability becomes area under a curve
Probability Distribution Function
The probability density function (pdf) maps a continuous variable to a probability density.
As the name "density" suggests, the area under the pdf curve between a range is the probability of the variable being in that range.
Total area under the curve must be , as chances of
events happening is 100% if the range includes all possible events.
There is no area under a single point
Mean and Variance
The mean and variance calculations are pretty much the same as that of discrete random variables, except the summations are swapped out for integrals.
Median and Percentile
The a-th percentileis the point at which a percent the area under the curve is to one side. You want to be a%, the calculation of which is in the page above.
By the same logic, the quartiles are at 25%, 50%, and 75% accordingly.
Uniform Distribution
Uniform random variable is described by two parameters: is minimum, and is maximum. It has a rectangular distribution, where every point has the same probability density.
CDF
Mean
Variance