3.3 - Continuous Probability Distributions

Overview Section

In the beginning of the course we looked at the difference between discrete and continuous data. The last section explored working with discrete data, specifically, the distributions of discrete data. In this lesson we're again looking at the distributions but now in terms of continuous data. Examples of continuous data include...

  • the amount of rainfall in inches in a year for a city.
  • the weight of a newborn baby.
  • the height of a randomly selected student.

Properties of Continuous Probability Functions Section

At the beginning of this lesson, you learned about probability functions for both discrete and continuous data. Recall that if the data is continuous the distribution is modeled using a probability density function ( or PDF).

We define the probability distribution function (PDF) of \(Y\) as \(f(y)\) where: \(P(a < Y < b)\) is the area under \(f(y)\) over the interval from \(a\) to \(b\). (see figure below)

The graph shows the area under the function f(y) shaded.
f(y) a b
Note! If Y is continuous \(P(Y = y) = 0\) for any given value y. Unlike the discrete random variables, the pdf of a continuous random variable does not equal to \(P(Y=y)\).

To find probabilities over an interval, such as \(P(a<Y<b)\), using the pdf would require calculus. Instead of doing the calculations by hand, we rely on software and tables to find these probabilities.

Expected value and Variance of a Continuous Random Variable Section

The expected value and the variance have the same meaning (but different equations) as they did for the discrete random variables.

Expected Value (or mean) of a Continuous Random Variable

The expected value (or mean) of a continuous random variable is denoted by \(\mu=E(Y)\).

Variance of a Continuous Random Variable

The variance of a continuous random variable is denoted by \(\sigma^2=\text{Var}(Y)\).

Standard Deviation of a Continuous Random Variable

The standard deviation of a continuous random variable is denoted by $\sigma=\sqrt{\text{Var}(Y)}$

Notice the equations are not provided for the three parameters above. Therefore, for the continuous case, you will not be asked to find these values by hand.

There are many commonly used continuous distributions. The most important one for this class is the normal distribution. We will describe other distributions briefly.