Continuous Random Variable (1)

Discrete Random Variables

• Probability Mass Function (PMF)

Continuous Random Variable

P[X=x]=0

Not possible to define a PMF for a continuous random variable

Discrete Random Variables

• Cumulative Distribution Function

PMF to CDF

Comparison

Discrete RV:1. Zero slope2. Jumps in CDF

Continuous RV:A continuous function

Slope of CDF function

The slope at any point x indicates the probability that X is near x.

Probability Density Function (PDF)

It is not possible to define a PMF function for a continuous variable because P[X=x]=0.We can, however, define a probability density function.

PDF of X

Example 3.3

Expected Value

Discrete Random Variable

Example

Find the expected stoppint point of the pointer

The Expected Value of a function

Derived Discrete Random Variable

Derived Continuous Random Variable

Discrete Example

Variance and Standard Deviation

Key Points

• An average is a typical value of a random variable.

• The next question:– “What are the chances of observing an

event far from the average?”• The variance of a random variable X describes

the difference between X and its expected value.

Definitions

Properties of Variance/Standard of Deviation

Discrete Example

Quiz 3.3