Continuous Uniform Distribution,
Normal Distribution
Continuous Uniform Distribution
A continuous random variable X which has probability density function given by:
 1
a xb
f ( x)   b  a
 0
x  [ a , b]
we write X ~  ( a , b)
Remember that the area under the graph of the random variable must be equal to 1 (see
continuous random variables).
Expectation and Variance
If X ~ U(a,b), then:

  E( X ) 
ab
2

Variance
Var( X ) 
(b  a ) 2
12
Proof of Expectation
image: http://revisionworld.com/sites/revisionworld.com/files/imce/uniform_expectation.GIF
Cumulative Distribution Function
The cumulative distribution function can be found by integrating the p.d.f between 0 and t:
Normal Distribution
For a finite population the mean (m) and standard deviation (s) provide a measure of
average value and degree of variation from the average value • If random samples of
size n are drawn from the population, then it can be shown (the Central Limit
Theorem) that the distribution of the sample means approximates that of a distribution
S
with mean   m and standard deviation  
.
n
The Probability Density Function pdf:

1
f ( x) 
e
 2
( x   )2
2 2
which is called normal distribution.
The pdf is characterized by its "bell-shaped" curve, typical of phenomena that
distribute symmetrically around the mean value in decreasing quantity as one moves
away from the mean
Empirical Rule for Normal Distribution:
The "empirical rule" is that
– approximately 68% of sample values are in the interval [μ-σ,μ+σ]
– approximately 95% are in the interval [μ-2σ,μ+2σ]
– almost all are in the interval [μ-3σ,μ+3σ]
This says that if n is large enough, then a sample mean for the population is accurate
with a high degree of confidence, since σ decreases with n What constitutes "large
enough" is largely a function of the underlying population distribution.
– The Central Limit Theorem assumes that the samples of size n which are used to
produce sample means are drawn in a random fashion.