Introduction to Probability Models

Lecture 22

Qi Wang, Department of Statistics

Oct 16, 2017

Reminders

  1. The third homework is due on NOW
  2. 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: $$P(a \le X \le b) = \int_{a}^{b} f(x) dx$$