Machine Learning - DISCo

which is wide enough to contain 95% of the total probability under this distribution.
This provides an interval surrounding errorv(h) into which errors(h) must
fall 95% of the time. Equivalently, it provides the size of the interval surrounding
errordh) into which errorv(h) must fall 95% of the time.
For a given value of N how can we find the size of the interval that contains
N% of the probability mass? Unfortunately, for the Binomial distribution
this calculation can be quite tedious. Fortunately, however, an easily calculated
and very good approximation can be found in most cases, based on the fact that
for sufficiently large sample sizes the Binomial distribution can be closely approximated
by the Normal distribution. The Normal distribution, summarized in
Table 5.4, is perhaps the most well-studied probability distribution in statistics.
As illustrated in Table 5.4, it is a bell-shaped distribution fully specified by its
Normal dismbution with mean 0. standard deviation I
3 -2 -1 0 1 2 3
A Normal distribution (also called a Gaussian distribution) is a bell-shaped distribution defined by
the probability density function
A Normal distribution is fully determined by two parameters in the above formula: p and a.
If the random variable X follows a normal distribution, then:
0 The probability that X will fall into the interval (a, 6) is given by
The expected, or mean value of X, E[X], is
The variance of X, Var(X), is
0 The standard deviation of X, ax, is
Var(X) = a2
ax = a
The Central Limit Theorem (Section 5.4.1) states that the sum of a large number of independent,
identically distributed random variables follows a distribution that is approximately Normal.
TABLE 5.4
The Normal or Gaussian distribution.