Answer:
• The normal distribution is the most important distribution. It describes well the
distribution of random variables that arise in practice, such as the heights or weights
of people, the total annual sales of a firm, exam scores etc. Also, it is important for the
central limit theorem, the approximation of other distributions such as the binomial,
etc.
• We say that a random variable X follows the normal distribution if the probability
density function of X is given by
f(x) = 1
σ
√
2π
e
− 1
2
(
x−µ
σ
)
2
, −∞ < x < ∞
This is a bell-shaped curve.
• We write X ∼ N(µ, σ). We read: X follows the normal distribution (or X is normally
distributed) with mean µ, and standard deviation σ.
• The normal distribution can be described completely by the two parameters µ and σ.
As always, the mean is the center of the distribution and the standard deviation is the
measure of the variation around the mean.
• Shape of the normal distribution. Suppose X ∼ N(5, 2).
x
f(x)
−3 −1 1 3 5 7 9 11 13
0.00 0.05 0.10 0.15 0.20
X ~ N(5,2)
• The area under the normal curve is 1 (100%).
Z ∞
−∞
1
σ
√
2π
e
− 1
2
(
x−µ
σ
)
2
dx = 1
• The normal distribution is symmetric about µ. Therefore, the area to the left of µ is
equal to the area to the right of µ (50% each).
1
• Useful rule (see figure above):
The interval µ ± 1σ covers the middle ∼ 68% of the distribution.
The interval µ ± 2σ covers the middle ∼ 95% of the distribution.
The interval µ ± 3σ covers the middle ∼ 100% of the distribution.
• Because the normal distribution is symmetric it follows that
P(X > µ + α) = P(X < µ − α)
• The normal distribution is a continuous distribution. Therefore,
P(X ≥ a) = P(X > a), because P(X = a) = 0. Why?
• How do we compute probabilities? Because the following integral has no closed form
solution
P(X > α) = Z ∞
α
1
σ
√
2π
e
− 1
2
(
x−µ
σ
)
2
dx = . . .
the computation of normal distribution probabilities can be done through the standard
normal distribution Z:
Z =
X − µ
σ
Theorem:
Let X ∼ N(µ, σ). Then Y = αX + β follows also the normal distribution as follows:
Y ∼ N(αµ + β, ασ)
Therefore, using this theorem we find that
Z ∼ N(0, 1)
Step-by-step explanation:
