Uniform Distribution
•f(x) =
1
b−aa≤x≤b
0 otherwise •F(x) =
0x≤a
x
b−aa<x≤b
1x>b
•E[X] = a+b
2•Var(X) = (b−a)2
12 •MX(t) = etb −eta
t(b−a)*
Uniform distributions are those with constant density function over an interval. The constant is equal to
the reciprocal of the length of the integral, since the density function must integrate to 1.
Discrete distributions can also have uniform probability functions, such as the one that describes the prob-
ability of each outcome when rolling a standard die. These distributions are not discussed much in practice.
*Note: Moment Generating Functions will be discussed in a later section, but I have provided the MGF for
reference. Some people recommend you memorize all MGFs for common random variables. I would suggest
only memorizing a couple of them: the MGF for an exponential and for general discrete random variables.
As we progress through the course, I will let you know if there are others worth memorizing.
Joint Uniform Distribution: f(x, y ) = 1
[AREA OF RE GI ON ]
This is the easiest way to characterize joint uniform distributions. If asked for the probability of a subset,
B, of a uniform region, you can set up a double integral:
ZZ
B
1
T otal Area dA =1
T otal Area ZZ
B
dA =Area B
T otal Ar ea
Since the density is constant, we can pull it out of the integration, and we find the probability of a region is
just the ratio of its area to the area of the sample space. Because of this result, probability question with
joint uniform densities are often worked geometrically, rather than with integrals.
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