Foundations of Data Science

The density of y is the unit variance, zero mean Gaussian, thus E(yy T ) = I.
Bernoulli trials and the binomial distribution
A Bernoulli trial has two possible outcomes, called success or failure, with probabilities
p and 1 − p, respectively. If there are n independent Bernoulli trials, the probability of
exactly k successes is given by the binomial distribution
( n
B (n, p) = p
k)
k (1 − p) n−k
The mean and variance of the binomial distribution B(n, p) are np and np(1 − p), respectively.
The mean of the binomial distribution is np, by linearity of expectations. The
variance is np(1 − p) since the variance of a sum of independent random variables is the
sum of their variances.
Let x 1 be the number of successes in n 1 trials and let x 2 be the number of successes
in n 2 trials. The probability distribution of the sum of the successes, x 1 + x 2 , is the same
as the distribution of x 1 + x 2 successes in n 1 + n 2 trials. Thus, B (n 1 , p) + B (n 2 , p) =
B (n 1 + n 2 , p).
When p is a constant, the expected degree of vertices in G (n, p) increases with n. For
example, in G ( n, 2) 1 , the expected degree of a vertex is n/2. In many real applications,
we will be concerned with G (n, p) where p = d/n, for d a constant; i.e., graphs whose
expected degree is a constant d independent of n. Holding d = np constant as n goes to
infinity, the binomial distribution
( n
Prob (k) = p
k)
k (1 − p) n−k
approaches the Poisson distribution
Prob(k) = (np)k e −np = dk
k! k! e−d .
To see this, assume k = o(n) and use the approximations n − k ∼ = n, ( )
n ∼= n k
( , and
k k!
1 −
1 n−k
∼
n)
= e −1 to approximate the binomial distribution by
( ( ) k n
lim
)p k (1 − p) n−k = nk d
(1 − d
n→∞ k
k! n n )n = dk
k! e−d .
Note that for p = d , where d is a constant independent of n, the probability of the binomial
distribution falls off rapidly for k > d, and is essentially zero for all but some
n
finite number of values of k. This justifies the k = o(n) assumption. Thus, the Poisson
distribution is a good approximation.
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