248
F. Firouzi et al.
Fig. 5.2 The left is uniform distribution over [−1,1]; the right is Gaussian distribution with
mean = 0 and variance = 1
In short, we can use an informal representation of the above equations:
p(x) := Pr (X = x)
If we take continuous random variable into account, the distribution function can be
rewritten as a probability density function (PDF). As in the case of a PMF, PDF must
be non-negative and sum to one. We use a similar distribution format as PMF and
illustrate with two distributions: the uniform distribution and Gaussian distribution,
respectively (Fig. 5.2).
p(x) =
1
b−a , if x ∈ [a, b]
0, otherwise
p(x) =
1
√
2πσ 2
exp
−
(x − μ)
2
2σ 2
5.1.2.3 Mean, Variance, and Covariance
Mean is defined as the average of the numbers: a calculated “central” value of a set
of numbers. Suppose we have an array of [1, 4, 6], we can simply calculate the mean
as (2 + 7 + 9)/3 = 6.
In statistics, we often need to know what the expected value of a random variable
is. For example, we may ask a question on what the expected temperature is
during a certain period of time. We also leverage the concept of “mean” to define
expectations and related quantities of distributions.