The uniform distribution stands out as a continuous probability distribution where every value within a specific range is equally likely to occur. This interval is defined by two constants, a and b, which can be either negative or positive. Unlike discrete distributions, continuous distributions require a different visual approach than histograms due to the infinite number of possible values. Instead, we use lines or curves to represent them, where areas under these graphical elements correspond to probabilities.
In the case of a uniform distribution, the defining characteristic is that all values across its interval (a, b) have an equal chance of appearing. This consistent probability leads to a distinctive graphical representation: a rectangle. Consider the uniform distribution illustrated in the interval (0, 10) as a prime example.
The horizontal axis of this graph represents the range of possible values for the variable X, in this case, from 0 to 10. Importantly, the uniform distribution assigns a probability of zero to any value of X that falls outside this specified interval.
The uniform distribution graph over the interval (0, 10) showcasing equal probability for all values within the range.
The width of this rectangular graph is determined by the interval’s range, calculated as the upper limit (b) minus the lower limit (a). For our example, this width is 10 – 0 = 10. This width effectively forms the base of our rectangle. The height of this rectangle is then calculated as 1 divided by the base, which in this case is 1/10. This specific calculation of height is crucial because it ensures that the total area of the rectangle always equals 1. Why is this important? Because in probability distributions, the total probability must always be 1, and graphically, this total probability is represented by the area under the curve or, in our case, under the rectangle.
Therefore, the area of the rectangle, calculated by multiplying the base times the height, perfectly represents the total probability of the uniform distribution, which is always equal to 1. Any area under this rectangle, within a sub-interval, will represent the probability of X falling within that specific sub-interval.
