Continuous Distributions Continued

Recall F_x(x) =

$$F_x(X)=1-e^{-\lambda x^{\beta }}$$

• A reminder! The median is the halfway point in a function right? (so would you agree that’s 0.5 probability?)

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Uniform Distribution

• The simplest continuous distribution
• All outcomes are equally likely

Uniform Distribution

• Now, you have to remember the bounds of the function. It’s given in each f(x), and is quite important. If it is outside of those bounds, it is zero.

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Example 2

• We are told that X is uniformly distributed between 0 & 1. (note, [0,1) means 0 is included, but 1 is not)

Now, for the variance of Y…

Reminder:

• The expected value of a constant is the constant,
• The variance of a constant is zero,
• Constants can be pulled out from of expected values
• Constants can be pulled out of variances, but must be squared

Normal Distribution

• Most popular distribution due to the Central Limit Theorem, which states that the sums of random variables tend to a normal distribution.

Normal (Gaussian) Distribution