Continuous Random Variable (1)Section 3.1-3.3

Continuous Random Variable

What is the probability that X is equal to x?

CDF for a Discrete Random Variables

Question: Is there a CDF for a continuous random variable if a PMF can not beidentified for a continuous random variable?

CDF for a Wheel-Spinning Experiment

P[X=≤ x]=x if 0 ≤X ≤1

CDF for Continuous Random Variable

• Even though it is not possible to define a PDF for a continuous random variable, it is possible to define a CDF for a random variable

PMF to CDF for a Discrete Random Variable

Theorem 2.2

Theorem 2.3

What contributes to the jump in the CDF?

Discrete RV

Continuous RV

Compare CDF of a Continuous RV to that of a Discrete RV

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.(Just as the jump in the CDF of a discrete RV suggests non-zero probability at X=x, so does a slope in CDF of a continuous random variable?)

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.

Properties of fX(x)

PDF of X

Expected ValueDiscrete Random Variable