Continuous Random Variable (1) Section 3.1-3.3. Continuous Random Variable What is the probability that X is equal to x?

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Presentation transcript:

Continuous Random Variable (1) Section

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 be identified 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 slope 2.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 f X (x)

PDF of X

Expected Value Discrete Random Variable