Continuous Random Variable: Difference between revisions

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The probability density function (pdf) maps a continuous variable to a
The probability density function (pdf) maps a continuous variable to a
probability density. As the name "density" suggests, the area under the
probability density.
pdf curve between a range is the probability of the variable being in that range.
 
As the name "density" suggests, the area under the pdf curve between a
range is the probability of the variable being in that range.


<math>
<math>
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<math>
<math>
\int_{-\infty}^\infty f(x) dx = 1
\int_{-\infty}^\infty f(x) dx = 1
</math>
There is no area under a single point
<math>
P(X = a) = 0
</math>
= 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.
<math>
E(X) = \mu_X = \int_{-\infty}^\infty x f(x) dx
</math>
<math>
Var(X) = \sigma^2_X = \int_{-\infty}^\infty (x - \mu_X)^2 f(x) dx
= \int_{-\infty}^\infty x^2 f(x) dx - \mu_X^2
</math>
</math>


= Uniform Distribution <math> X \sim Uniform(a, b) </math> =
= Uniform Distribution <math> X \sim Uniform(a, b) </math> =
<math> a </math> is minimum, and <math> b </math>
<math> a </math> is minimum, and <math> b </math>

Revision as of 07:38, 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.

Uniform Distribution

is minimum, and