Single Continuous Random Variable

Definition

A continuous random variable is a random variable which can take any real value in an interval. For continuous random variables . Instead probabilities are defined over an interval e.g for we consider .

PDF (Probability Density Function)

In order to do this we have a probability density Function which satisfies

3. as total probability sums to

CDF (Cumulative Density Function)

A Cumulative Density Function is defined as

It has the properties that

1. is non-decreasing
4. If is Differentiable then

Expected Value

The Expected Value denoted of is the mean value of . It is defined/found as

This comes from a generalisation of the discrete case . We can also compute the expected value of any function

Also note that since is an Integral it is linear

Variance and Standard Deviation

Variance is defined as

Since is the mean, is the deviation of from the mean. So variance is the mean squared distance of from the mean. It is a meauser of Spread of the variable/data.

It has the properties that:

1. .
2. Weight larger deviations more than small ones. Hence the squaring.

We then define standard deviation as

To go from squared distance to distance to get spread in the original units. However it is NOT quite the average distance. But we use over as the standard deviation has some nicer properties.