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Definitions and properties of continuous random variables, probability density functions, mean, variance, skewness, and kurtosis. It also introduces the normal distribution and its moments, as well as the standard normal distribution and its properties. Central limit theorem (lindberg-levy) and its application to the normal approximation of binomial distributions are also discussed.
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Def
n of a continuous random variable
Consider an experiment having an infinite number of
possible outcomes.
Assign some real number to each possible outcome,
and associate this number with some random variable,
say X.
Example: Weighing jelly-beans
Imagine purchasing a bag of 100 jelly beans, and
weighing each bean on a scale that is very accurate
and very precise.
X = weight of jellybean
Def
n of the probability density function of a continuous
random variable X
Let X be a continuous random variable, assuming all
values in some range Rx^ of the real line.
Then the probability density function of X is that
function f(x) such that
b
a
P a ≤ X ≤ b = f x dx ) ∫
Properties of f ( ) : x
(1) f ( x ) ≥ 0 for all x
(2) f ( ) x dx 1
∞
−∞
∫
Def
n : Skewness of a continuous random variable X
3 3 x^ f^ ( ) x^ dx
∞
−∞
μ = − μ ∫
Def
n : Kurtosis of a continuous random variable X
4
4 x^ f^ ( ) x dx
∞
−∞
μ = − μ ∫
Def
n : Cumulative Distribution Function of a continuous
random variable X
x
F x P X x
f y dy −∞
∫
Def
n : The normal d
n
A continuous random variable X is said to have a
normal distribution with parameters μ and
if the probability density
function of X is
2 σ
2
( )
2
2
1 (^12) ( ) , 2
x
f x e x
−μ − σ = − πσ
Notation:
2 X ~ N ( ,μ σ )
Moments:
Mean : E X ( ) = μ
Variance :
2 Var X ( ) = σ
Skewness :
2 3 (^3 1 ) 2
μ μ = → β = =
σ
Kurtosis :
4 2 4 3 2 0
μ β = − → β = σ
Def
n : The standard normal d
n
2
2
1 2
−
μ = σ =
π
x f x e ∞ < x < ∞
Notation: X ~ N (0,1)
Property:
If
2 ~ ( , ), and if ,
− μ μ σ = σ
X N z
then z ~ N (0,1)
The Standard Normal Distribution
Standard Normal Distribution
-4 -3 -2 -1 0 1 2 3 4
z
P[z>2.37]=.
Standard Normal Distribution
-4 -3 -2 -1 0 1 2 3 4 z
.
.
z=2.
Reproductive Property:
Let independent normally distributed
random variables, with
x 1 (^) , x 2 , L, xn be n
2 ~ ( , i xi i X N μ σ x )
Let
1
n
i i i
y a
=
= (^) ∑ X
Then 2 y ~ N ( μ (^) y ,σ y )
where
1
2 2
1
i
i
n
y i i
n
y i i
a
a
=
=
2
x
x
μ = μ
σ = σ
∑
∑
Application : Normal Approximation to Binomial D
n
Let X (^) 1 , L, X (^) n be iid Bernoulli random Variables, so
that
1
=
= (^) ∑
n
i i
Y X Binomial n , p )
with
2
= μ =
= σ = −
y
y
E Y np
Var Y np p
Then
(0,1) as (1 )
− μ (^) − = = → → σ (^) −
y
y
Y (^) Y np Z N np p
n ∞
Rule of Thumb :
Normal Approximation to binomial is good if
min( np nq , ) ≥ 5
-. -. -. -. -. -. -. -. -. -. -. -. -. -. -. -. -. -. -. -.
Using normal approximation:
0.1806 not so good.
X np P X P np p
0
P r o p o r t i o n x
Using normal approximation with continuity
correction:
0.3242 much better.
X np
P X P np p
0
P r o p o r t i o n x
