The Normal Distribution
The normal or bell shaped distribution is the most widely used distribution.
Definition:
A random variable 𝑌 has a normal distribution if and only if
𝑓(𝑦) =
1
2 /(2𝜎 2 )
𝜎√2𝜋
𝑒 −(𝑦−𝜇)
−∞ < 𝑦 < ∞ and − ∞ < 𝜇 < ∞, 𝜎 > 0.
Theorem:
If 𝑌 has a normal distribution, then 𝐸(𝑌) = 𝜇 and 𝑉(𝑌) = 𝜎 2 .
If 𝑌 has a normal distribution with mean 0 and standard deviation 1, then
𝑓 (𝑦) =
1
√2𝜋
𝑒 −𝑦
2 /2
−∞ < 𝑦 < ∞
In this case, 𝑌 is a standard normal distribution and is denoted by 𝑍.
The probability of a normal distribution is on page 848.
1
2
Example 4.8
Let Z denote a normal random variable with mean 0 and standard deviation 1.
a. Find 𝑃(𝑍 > 2) (Answer is 0.0228)
b. Find 𝑃(−2 ≤ 𝑍 ≤ 2) (Answer is 0.9544)
c. Find 𝑃(0 ≤ 𝑍 ≤ 1.73) (Answer is 0.4582)
Example 4.9
The achievement scores for a college entrance examination are normally
distributed with mean 75 and standard deviation 10. What fraction of the
scores lies between 80 and 90?
𝑦1 − 𝜇 80 − 75
𝑧1 =
=
= 0.5
𝜎
10
𝑦2 − 𝜇 90 − 75
𝑧2 =
=
= 1.5
𝜎
10
𝐴 = 𝐴0.5 − 𝐴1.5 = 0.3085 − 0.0668 = 0.2417.
3