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Integrate uniform date

hey guys it's mark from ace tutors and over the course of the next several videos i'm going to explain some of the most common probability distributions covered in statistics in this video i'll go over the uniform distribution so first off a uniform distribution is characterized by having a constant probability within a certain domain but what does this mean well to find out let's take a look at this distribution on a graph where the x-axis represents the different values this certain event can be well the y-axis represents the probability of those events from this graph you can hopefully see how each x-value within the domain of a to b has the exact same y value which implies the same exact probability but what is this constant probability well we already know that if we add up all the probabilities of all the outcomes of a certain event or in other words find the total area under any probability distribution the result must equal 1. since this area is a nice and simple rectangle we can find its area by multiplying its height by its width the width is simply found by subtracting b minus a and now we can solve this equation for what the height must be dividing both sides by b minus a we get that the height of this distribution must be one over b minus a okay now that we covered the general look of the uniform distribution let's go over some common values associated with this distribution first let's find its mean or expected value because the distribution is nice and symmetric the mean mu ends up being right in the middle of its domain which you can find by taking the average of the domain bounds aka a plus b divided by two and once again from the distribution symmetry this value ends up being the same for the median okay moving on to standard deviation this formula ends up being a little less intuitive for a uniform distribution the standard deviation sigma ends up being the square root of b minus a squared divided by 12. next let's go over some of the fun stuff probability when we discuss finding probability for continuous functions like this we generally want to find the probability of a range of things occurring for example we might want to find the probability of our event's outcome being between c and d to calculate this we could approach it one of two ways you can either see it as a rectangle and calculate the area by length times width or understanding that the distribution is uniform you can find the fraction that the desired interval makes up the entire distribution's domain either way you would end up with the same result of d minus c divided by b minus a now that we covered some general terms from the uniform distribution let's put the theoretical garbage aside and see how this works with a...