IEE 380 Ch 4 Continuous Uniform Distribution

well you have seen this distribution before on a previous slide it is called The Continuous uniform distribution and it has this form so whenever you see a probability density function that has the form 1 over B minus a and then X is allowed to be between A and B then you have something

called The Continuous uniform distribution remember we could also put equal signs here or we could go less than or less than or equal to or we could go less than or equal to or less than because the probability that X which is clearly a continuous distribution um the probability equals anything is equal to zero so if we apply the formula

to get the expected value or the mean of X it turns out that it works out to this very nice little formula called a plus b over 2. same thing for the variance if I apply the formula the expected value of x squared minus the mean squared I'm going to get B minus a squared over 12

and if I take the square root of that I'm going to get the standard deviation now to get the CDF of X what we're going to do is um what we do is we go from well first of all it's going to be zero whenever X is less than a and remember what capital f of x is capital f of x is

the probability that my random variable is less than or equal to X and for a continuous random variable it could be just strictly less than x so whenever X is less than a I don't have any probability there so the probability X is less than anything less than a has to be zero we also know that

it will equal 1 whenever X is bigger than b or bigger than or equal to B because if I were to put my lawnmower right here and then I want to mow to the left and I said and that would pick up all of the probability that X is any for any X bigger than b make it B plus one