Foundations of Data Science

a probability distribution function.
In many situations, a probability distribution function does not exist. For example,
for the uniform probability on the interval [0,1], the probability of any specific value is
zero. What we can do is define a probability density function p(x) such that
Prob(a < x < b) =
∫ b
a
p(x)dx
If x is a continuous random variable for which a density function exists, then the cumulative
distribution function f (a) is defined by
f(a) =
which gives the probability that x ≤ a.
∫ a
−∞
p(x)dx
12.4.1 Sample Space, Events, Independence
There may be more than one relevant random variable in a situation. For example, if
one tosses n coins, there are n random variables, x 1 , x 2 , . . . , x n , taking on values 0 and 1,
a 1 for heads and a 0 for tails. The set of possible outcomes, the sample space, is {0, 1} n .
An event is a subset of the sample space. The event of an odd number of heads, consists
of all elements of {0, 1} n with an odd number of 1’s.
Let A and B be two events. The joint occurrence of the two events is denoted by
(A∧B). The conditional probability of event A given that event B has occurred is denoted
by Prob(A|B)and is given by
Prob(A|B) =
Prob(A ∧ B)
.
Prob(B)
Events A and B are independent if the occurrence of one event has no influence on the
probability of the other. That is, Prob(A|B) = Prob(A) or equivalently, Prob(A ∧ B) =
Prob(A)Prob(B). Two random variables x and y are independent if for every possible set
A of values for x and every possible set B of values for y, the events x in A and y in B
are independent.
A collection of n random variables x 1 , x 2 , . . . , x n
possible sets A 1 , A 2 , . . . , A n of values of x 1 , x 2 , . . . , x n ,
is mutually independent if for all
Prob(x 1 ∈ A 1 , x 2 ∈ A 2 , . . . , x n ∈ A n ) = Prob(x 1 ∈ A 1 )Prob(x 2 ∈ A 2 ) · · · Prob(x n ∈ A n ).
If the random variables are discrete, it would suffice to say that for any real numbers
a 1 , a 2 , . . . , a n
Prob(x 1 = a 1 , x 2 = a 2 , . . . , x n = a n ) = Prob(x 1 = a 1 )Prob(x 2 = a 2 ) · · · Prob(x n = a n ).
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