2- The Normal random variables
We say that X is a normal random variable, or simply that X is
normally distributed, with parameters μ and 
if the density of X is
given by
The normal distribution was introduced by the French mathematician
Abraham DeMoivre in 1733, who used it to approximate probabilities
associated with binomial random variables when the binomial
parameter n is large.
Note that f (x) is indeed a probability density function, so
An important fact about normal random variables is that if X is normally
distributed with parameters μ and 
, then Y = a X + b is normally
distributed with parameters a μ + b and
. To prove this statement,
suppose that a > 0. (The proof when a < 0 is similar.) Let πΉΰ― denote the
cumulative distribution function of Y. Then
where 
is the cumulative distribution function of X. By
differentiation, the density function of Y is then
which shows that Y is normal with parameters a μ + b and
An important implication of the preceding result is that if X is normally
distributed with parameters μ and 
, then 
is normally
distributed with parameters 0 and 1. Such a random variable is said to
be a standard, or a unit, normal random variable.
We now show that the parameters μ and
of a normal random variable
represent, respectively, its expected value and variance.
EXAMPLE 10:
Find E(X) and Var(X) when X is a normal random variable with
parameters μ and
Solution.
Let us start by finding the mean and variance of the standard normal
random variable
.We have
Thus,
Integration by parts (with u = x and dv =π₯ 
now gives
Because X = μ + σ Z, the preceding yields the results
It is customary to denote the cumulative distribution function of a
standard normal random variable by π·(π₯). That is,
For negative values of x, π·(π₯) can be obtained from the relationship
π·(−π₯) = 1 − π·(π₯) − ∞ < π₯ < ∞
This equation states that if Z is a standard normal random variable,
then
π(π ≤ −π₯) = π(π > π₯) − ∞ < π₯ < ∞
Since π = 
is a standard normal random variable whenever X is
normally distributed with parameters μ and 
, it follows that the
distribution function of X can be expressed as
The values of π·(π₯) for nonnegative x are given in the following Table.
EXAMPLE 11:
If X is a normal random variable with parameters μ = 3 and 
= 9,
find (a) P{2 < X < 5}; (b) P{X > 0}; (c) P{|X − 3| > 6}.
Solution.
(a)
(b)
(c)
* The Normal Approximation to the Binomial Distribution
An important result in probability theory known as the DeMoivre–
Laplace limit theorem states that when n is large, a binomial random
variable with parameters n and p will have approximately the same
distribution as a normal random variable with the same mean and
variance as the binomial. This result was proved originally for the
special case of p = 
by DeMoivre in 1733 and was then extended to
general p by Laplace in 1812. It formally states that if we “standardize”
the binomial by first subtracting its mean np and then dividing the result
by its standard deviation 
, then the distribution function of
this standardized random variable (which has mean 0 and variance 1)
will converge to the standard normal distribution function as n → ∞.
Note that we now have two possible approximations to binomial
probabilities: the Poisson approximation, which is good when n is
large and p is small, and the normal approximation, which can be
shown to be quite good when np(1 − p) is large. (The normal
approximation will, in general, be quite good for values of n satisfying
ππ (1 − π) ≥ 10).
EXAMPLE 12:
Let X be the number of times that a fair coin that is flipped 40 times lands on heads. Find the probability that X = 20.
Use the normal approximation and then compare it with the exact solution.
Solution.
To employ the normal approximation, note that because the binomial is
a discrete integer-valued random variable, whereas the normal is a
continuous random variable, it is best to write π(π = π) as
before applying the normal approximation (this is
called the continuity correction). Doing so gives
The exact result is
