An Introduction to Continuous Probability Distributions

Let's look at an introduction to continuous random variables and continuous probability distributions. Continuous random variables can take on an infinite number of possible values, corresponding to every value in an interval. So for example we might have a random variable that takes on any value between 3 and 4. Our random variable might take on the value 3.1, or 3.12, or 3.1278694, or 3.8, or what have you. Any of the infinite number of values in between 3 and 4. Or a common one that we see, we have a random variable taking on any positive value, so any value between 0 and infinity Here is approximately the distribution of the height of adult Canadian males. Now height is a continuous random variable and it's going to have a continuous probability distribution. And this looks something like a smooth version of a histogram. A little loosely speaking, values of the variable where the curve is high are more likely to occur than where it is low. So here we would be more likely to get a height in this range than way out here in this extreme. We cannot model continuous random variables with the same methods we used for discrete random variables. There will be some similarities, but we will have to use different methods. We model a continuous random variable with a curve, f(x), called a probability density function or pdf. Here's another example of a continuous probability distribution, the distribution of time to failure in thousands of hours, for a type of light bulb. Values the random variable can take on are given down here on the x axis, and the probability density function f(x) is a function giving the height of the curve at those values of x. f(x) represents the height of the curve at point x. An important notion is that for continuous random variables, probabilities are areas under the curve. Here's a continuous probability distribution for a random variable X. And the height of the curve is represented by f(x), and the probability the random variable X falls in between two values a and b, is simply the area under the curve between a and b. One important notion here is that the probability the random variable X is exactly equal to any one specific value is 0. We could say the probability the random variable X is equal to the value a is 0 for any a. We could think of the point a here as an infinitesimally small point with infinitesimally small area above it we call that area 0, So the probability that X is equal to a for any a is 0. So for any continuous probability distribution, let's say the probability that X is equal to 3.12, that's going to be equal to 0. So from a practical point of view it's only going to make sense to talk about the random variable X falling in an interval values. One implication of what we just talked about here is that this probability would be the same as saying the probability that the random variable X is greater than or equal to a and less than or equal to b. We can switch less than or equal to with less than, it doesn't matter because the probability the random variable X is exactly equal to one specific value is 0. For any continuous probability distribution, f(x) has to be at least 0 everywhere. Note that there is no upper bound on it, it can take on values greater than 1. One restriction is that the area under the entire curve is equal to 1. And so these two restrictions ensure that all probabilities lie between 0 and 1, and the probability of something happening is 1. There are a number of common continuous probability distributions that come up frequently in theory and practice. One very common and extremely important continuous probability distribution is the normal distribution. And it looks like this. This is the continuous uniform distribution for which f(x) is constant over the range of possible values of x. The exponential distribution look something like this. This is something we might see in exponential decay or a number of other spots. And there are many other continuous probability distributions that are very important to us in probability and statistics. Probabilities and percentiles are found by integrating the probability density function. Probabilities are areas under the curve, and areas under the curve are found using integration. Fortunately for us statistical software will carry out the integration for us in a lot of situations. And so in practice will be using statistical software or statistical tables to find these areas. Deriving the mean and variance of the probability distribution also requires integration. Depending on your needs you may or may not need to know how to actually carry out the integration, so I'm going to look at those concepts in separate videos.

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Keywords: continuous random variables, continuous, random variables, probability distributions, probability density function, pdf, f(x), normal distribution, uniform distribution, introductory statistics, introductory, statistics, jbstatistics, jb statistics, 8msl, 8 minute stats lectures, intro stats videos, intro stats help, stats help, stats tutor, jeremy balka, AP statistics

Id: OWSOhpS00_s

Length: 5min 51sec (351 seconds)

Published: Sun Dec 23 2012