Download Chapter 6: Normal Distribution

Page 1 of 7
Chapter 6: Normal Distribution
•
Section 6.1 Continuous Random Variables
Measurements of weight, height and temperature are represented by continuous random
variables.
Definition: f(x) is called a probability density function of a continuous random variable X if
(1)
∫
∞
−∞
f ( x)dx = 1
(2) f ( x) ≥ 0
For continuous random variable X
•
P(a ≤ X ≤ b) =
•
P(X = b) = 0
∫
b
a
f ( x)dx = area under the density curve between a and b.
f(x)
a
b
x
P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a < X < b)
Because of symmetry P(Z > 0) = P(Z < 0) = 0.5 and P(Z < − a) = P(Z > a)
Page 2 of 7
•
Section 6.2 – 6.3 Normal Probability Distribution
Definition: A random variable X is said to have a normal distribution if its probability density
function is given by
f ( x) =
1
σ 2π
e
1  x−µ 
− 

2 σ 
2
, for −∞ < x < ∞ .
The mean and variance of X are µ and σ 2 . They both characterize the normal density.
The above is bell - shaped or mound – shaped.
Definition: A random variable Z is said have a standard normal distribution if its probability
density function is given by
f ( z) =
1
2π
e− z
2
/2
, for − ∞ < z < ∞ .
The mean and variance of Z are µ = 0 and σ 2 = 1.
Note that Z =
X −µ
σ
.
Page 3 of 7
P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a < X < b)
A table exists for the computation of the above integral (see the inside cover of our textbook)
To find any (standard normal) probability:
•
Sketch the normal curve
•
Shade the required area
•
Use the table accordingly
Example:
a) P( Z < 1.5 )
b) P( Z > -2.42)
c) P( Z > 3.02)
d) P( Z >-2.33)
e) P( -2.50 < Z < -1.17)
f) ( -1.20 < Z < 2.40)
Page 4 of 7
Computing Percentiles
Example: Find b so that
a) P( Z < b ) = 0.9803
b) P( Z > b) = 0.9960
c) P( Z < b) = 0.0166
d) P( Z > b ) = 0.0017
Suppose we want to find the probability of any random variable X
X −µ
•
Standardized the random variable X to obtain Z =
•
Follow previous method to find required probability.
σ
Example: If X is normally distributed with µ = 38 and σ = 5, find:
a) P( X > 46 )
b) P( X < 36)
Page 5 of 7
c) P( 31 < X < 41)
Examples
Ex 6.11 page 225
Ex 6.26 page 227
Ex 6.32 page 228
Page 6 of 7
Supplementary #10, #11, #12
Page 7 of 7
•
Section 6.4 Normal Approximation to the Binomial (optional)
Recall that a binomial random variable X has mean µ = np and standard deviation σ = npq . If n
for a binomial distribution is large and p is not too close to 0 or 1, we may use normal
approximation to the binomial.
X − np
npq
•
We standardized X to obtain Z =
•
We compute P(X ≥ a) as in the normal distribution
Note: Since the original binomial random variable is discrete and we want to use a continuous
normal distribution to approximate it, we apply a continuity correction. Continuity correction is
the addition or subtraction of 0.5.
Binomial
Normal Approximation
P(X ≤ b)
P(X ≤ b + 0.5)
P(X ≥ a)
P(X ≥ a – 0.5)
P(a ≤ X ≤ b)
P(a – 0.5 ≤ X ≤ b + 0.5)
P(X = a)
P(a – 0.5 ≤ X ≤ a + 0.5)
For any other inequality < (or >), first change < to ≤ (or > to ≥ ). You may add to the given
probability, but do not take away.
For a binomial random variable X, P(X < 4) = P(X ≤ 3) and P(X > 7) = P(X ≥ 8).
Example: Ex 6.44 page 235