Continuous Random Variable: Difference between revisions

From Rice Wiki
No edit summary
 
(16 intermediate revisions by 2 users not shown)
Line 1: Line 1:
[[Category:Distribution (Statistics)]]
Continuous random variables have an inifinite number of values for any
Continuous random variables have an inifinite number of values for any
given interval. While similar, the approach to analysis is very
given interval. While similar, the approach to analysis is very
Line 5: Line 6:
* Probability becomes area under a curve
* Probability becomes area under a curve


= Probability Distribution Function <math> f(x) </math> =
= Probability Density Function =


The probability density function (pdf) maps a continuous variable to a
The probability density function (pdf) maps a continuous variable to a
Line 34: Line 35:


The mean and variance calculations are pretty much the same as that of
The mean and variance calculations are pretty much the same as that of
discrete random variables, except the summations are swapped out for
[[Discrete Random Variable|discrete random variables]], except the summations are swapped out for
integrals.
integrals.


Line 90: Line 91:
</math>
</math>


= Exponential Distribution <math>X \sim Exp(\lambda)</math> =
= Exponential Distribution =


The exponential distribution models events that occurs
The exponential distribution models events that occurs
Line 97: Line 98:
* At a constant average rate
* At a constant average rate


It takes in one parameter: <math>\lambda</math>, the rate parameter. It
It takes in one parameter: <math>\lambda</math>, the '''rate parameter.''' Defined by the mean below, it is the ''average rate per unit time/space.''
is defined by the mean below.


Exponential distribution has the '''memoryless property''': the
Exponential distribution has the '''memoryless property''': the
Line 112: Line 112:
</math>
</math>


Notably, it models time until some event has happened.
Notably, it models time until some event has happened, in contrast to [[Discrete Random Variable#Poisson|poisson distribution]], which measures the number of events in a unit time.


==== PDF ====
==== PDF ====
Line 152: Line 152:


= Normal Random Variable =
= Normal Random Variable =
 
[[File:Z score table.png|thumb|Z score table]]
Normal random variables are the most widly used continuous RV in
'''Normal random variables''' (aka. Gaussian RV) are the most widly used continuous RV in
statistics, characterizing many natural phenomenons. It is the famous
statistics, characterizing many natural phenomenons. It is the famous
bell curve.
bell curve.
They are characterized by two parameters: mean and variance.


<math>
<math>
X \sim N(\mu_X, \sigma^2_X)
Y \sim N(\mu_Y, \sigma^2_Y)
</math>
 
Normal random variables are perfectly symmetric at the mean.
 
==== Standardizing Normal Distribution ====
 
Standardization of a data means to make its mean 0 and its standard
deviation 1. We do this by subtracting the mean and dividing by the
standard deviation:
 
<math>
Z = \frac{Y - \mu}{\sigma}
</math>
 
Intuitively, this moves the dataset and changes the scale. We do this to
simplify probability calculations.
 
==== Z score ====
 
The z-score is the number of standard deviations above or below the
mean. A positive z score is above, and a negative is below.
 
<math>
z = \frac{y - \mu}{\sigma}
</math>
</math>


==== PDF ====
==== PDF ====
The pdf for normal random variable is the following.


<math>
<math>
f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2} \frac{(x -
f(y) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2} \frac{(y -
\mu)^2}{\sigma^2}}
\mu)^2}{\sigma^2}}
</math>
</math>


==== CDF ====
After standardizing the normal RV, we can use the following instead.


<math>
<math>
F(x) = 1 - e^{- \lambda x}
f(y) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2} z^2}
</math>
</math>


==== Mean ====
where <math>z</math> is the z-score covered in the last section.
 
Integration by parts


<math>
<math>
\mu_X = \frac{1}{\lambda}
f(z) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} z^2}
</math>
</math>


==== Variance ====
==== Quantiles ====
 
Quantiles are points dividing the range of a probability
distribution. Quartiles and precentiles are types of quantiles.
 
For normal distributions, there are special points (critical values)
that correspond to particular probabilities: <math>z_a</math>, where
<math>a</math> is the probability in the right tail.
 
==== Standard Normal Table ====
 
The standard normal table calculate lower tail values based on the
standard normal distribution (i.e. area under the curve left of the
point).
 
==== Linear Combinations of Independent Normal RV ====


Integration by parts
<math>
W = aX + bY
</math>


<math>
<math>
\sigma^2 = \frac{ 1 }{ \lambda^2 }
W \sim N(a\mu)X + b\mu_y, a^2 \sigma^2_X + b^2 \sigma^2_y)
</math>
</math>
= Other distributions =
[[Two Numerical RVs]]
[[Category:Statistics]]

Latest revision as of 04:08, 25 April 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 Density 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.

PDF

CDF

Mean

Variance

Exponential Distribution

The exponential distribution models events that occurs

  • Continuously
  • Independently
  • At a constant average rate

It takes in one parameter: , the rate parameter. Defined by the mean below, it is the average rate per unit time/space.

Exponential distribution has the memoryless property: the probability to an event does not change no matter how much time has passed.

In probability terms, the probability that we must wait an additional units given that we have waited units

Notably, it models time until some event has happened, in contrast to poisson distribution, which measures the number of events in a unit time.

PDF

CDF

Mean

Integration by parts

Variance

Integration by parts

Exponential and Poisson

Exponential distribution and poisson RVs are related:

  • : the number of events in a unit time
  • : waiting time until an event

Normal Random Variable

Z score table

Normal random variables (aka. Gaussian RV) are the most widly used continuous RV in statistics, characterizing many natural phenomenons. It is the famous bell curve.

They are characterized by two parameters: mean and variance.

Normal random variables are perfectly symmetric at the mean.

Standardizing Normal Distribution

Standardization of a data means to make its mean 0 and its standard deviation 1. We do this by subtracting the mean and dividing by the standard deviation:

Intuitively, this moves the dataset and changes the scale. We do this to simplify probability calculations.

Z score

The z-score is the number of standard deviations above or below the mean. A positive z score is above, and a negative is below.

PDF

The pdf for normal random variable is the following.

After standardizing the normal RV, we can use the following instead.

where is the z-score covered in the last section.

Quantiles

Quantiles are points dividing the range of a probability distribution. Quartiles and precentiles are types of quantiles.

For normal distributions, there are special points (critical values) that correspond to particular probabilities: , where is the probability in the right tail.

Standard Normal Table

The standard normal table calculate lower tail values based on the standard normal distribution (i.e. area under the curve left of the point).

Linear Combinations of Independent Normal RV

Other distributions

Two Numerical RVs