## Introduction to Probability Models

Lecture 22

Qi Wang, Department of Statistics

Oct 16, 2017

## Reminders

- The third homework is due on
**NOW**
- The fifth quiz will be at this
**Wednesday, Oct 18**

## Probability Mass Function Revisit

**Definition:**a function that gives the probability that a **discrete** random variable is exactly equal to some value.
- For every x, $0 \le p_X(x) \le 1$
- $\sum_x{p_X(x)} = 1$
- For example, $X \sim Binomial(n = 4, p = 0.3)$

## Continuous Random Variable

**Definition:**a random variable that can take on any value in a range.
- For example:
- Blood pressure
- The height of a 8 year old
- Mile per gallon of a car
- GPA

## Probability Density Function

In order to find probabilities of continuous random variables, we can no longer use a PMF, because the probabilities are no longer at points,
they are over regions. Instead we have a Probability Density Function, or PDF, $f(x)$, which looks more like a traditional function over a region.
If you are given a PDF, the probability can be calculated as:
$$P(a \le X \le b) = \int_{a}^{b} f(x) dx$$