Continuous Random Variables

1
Continuous Random Variables

• Chap. 12

2
Preamble

• Continuous probability distribution are not
  related to specific experiments
• There are multiple shapes (with parameters)
  that fulfill the probability distribution
  condition (area under shape equal to one.
• In general, you plot your sample data and try to
  find a shape that fits sample data.

3
Uniform Distribution
a
b
4
Standard Normal Distribution(One of most
important..)

• ? is the mean
• ? is the standard deviation

m
?
5
Standard Normal Deviation(How to use)

• Standard integration techniques cannot be used
• Tables exist for
• For any other random variable X with mean ? and
  standard deviation ?, we can use random variable
  Z such that

6
Gamma Distributions

• Standard Normal distribution is symmetric
• Need of skewed p.d.f
• Preliminary Gamma function ?(?) is defined
• P.d.f of a gamma distribution is defined as

7
Gamma Distributions (2)Probability Density
Function
k ?
? ?
8
Weibull Distribution

• Heavily used in fault-tolerance, reliability
  lifetime, time between failures
• Weibull with parameters a and g

9
Standard Weibull Distribution(a 1)
10
Measures of Central Tendency

• Mean (arithmetic, geometric, harmonique)
• Median
• Mode

11
Probability Plots

• Used to identify probability distribution of a
  sample data
• Building histograms with sample data is not
  always precise (e.g., small sample)
• Rather, build a probability plot based on pairs
  (X,Y) (nsample size)
• X 100 (i - 0.5)/nth percentile
• Y ith sample

12
Normal Probability Plot

• If the underlying probability distribution is a
  normal distribution
• Plot is a straight line
• With slope ??(Standard deviation)
• With intercept ???Mean)

13
Sum of Random Variables

• Given a linear combination of n random variables
• Y a1X1.. anXn
• E(Y) a1E(X1) .. anE(Xn)
• V(Y) a1?12 .. an?n2

14
Average of Random Variables(Corollary)

• Let Xi a random variable with mean ? and standard
  deviation ?.
• Let ai 1/n
• Y (X1..Xn)/n
• E(Y) ?
• V(Y) ?

15
Central Limit Theorem

• Let X1, X2,,Xn be a random sample from a
  distribution with mean ? and variance ?2. If n is
  sufficiently large, the average of these random
  variables has approximately a normal distribution
  with mean ? and variance ?2/n.
• The larger n, the better the approximation