Download Chapter 4, part II: Random Variable

Since everything is a reflection of our minds,
everything can be changed by our minds.
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Random Variables
Section
4.6-4.14
Types of random variables
Binomial and Normal distributions
Sampling distributions and Central
limit theorem
Random sampling
Normal probability plot
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What Is a Random Variable?
A random variable (r.v.) assigns a number to
each outcome of a random circumstance.
Eg. Flip two coins: the # of heads
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When an individual is randomly selected and
observed from a population, the observed
value (of a variable) is a random variable.
Types of Random Variables
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
A continuous random variable can take any
value in one or more intervals. We cannot list
down (so uncountable) all possible values of
a continuous random variable.

All possible values of a discrete random
variable can be listed down (so countable).
Distribution of a Discrete R.V.
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X = a discrete r.v.
x = a number X can take
The probability distribution of X is:
P(x) = P(Y=x)
How to Find P(x)
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P(x) = P(X=x) = the sum of the probabilities
for all outcomes for which X=x
Example:
toss a coin 3 times
and x= # of heads
Expected Value (Mean)
The expected value of X is the mean (average) value
from an infinite # of observations of X.
= a discrete r.v. ; { x1, x2, …} = all possible X values
pi is the probability X = xi where i = 1, 2, …
The expected value of X is:
X
  E ( X )   xi pi
i
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Variance & Standard Deviation

Variance of X:
V ( X )   2   ( xi   ) 2 pi

i
Standard deviation (sd) of X:

2
(
x


)
pi
 i
i
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Binomial Random Variables
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Binomial experiments (analog: flip a coin n
times):
Repeat the identical trial of two possible
outcomes (success or failure) n times
independently
The # of successes out of the n trials
(analog: # of heads) is called a binomial
random variable
Example
Is it a binomial experiment?
 Flip a coin 2 times
 The # of defective memory
chips of 50 chips
 The # of children
with colds in a family of 3 children
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Binomial Distribution
p = the probability of success in a trial
 n = the # of trials repeated independently
 Y = the # of successes in the n trials
For y = 0, 1, 2, …,n,

n!
y
n y
P(y) = P(Y=y)=
p (1  p )
y!(n  y )!
Where n! n(n  1)( n  2)...1
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Example: Pass or Fail
Suppose that for some reason, you are not
prepared at all for the today’s quiz. (The quiz is
made of 5 multiple-choice questions; each
has 4 choices and counts 20 points.)
You are therefore forced to answer these
questions by guessing. What is the probability
that you will pass the quiz (at least 60)?
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Mean & Variance of a Binomial R.V.

Notations as before

Mean is
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Variance is
  np
  np (1  p )
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Distribution of a Continuous R.V.

The probability distribution for a continuous
r.v. Y is a curve such that
P(a < Y <b) = the area under the curve
over the interval (a,b).
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Normal Distribution
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
The most common distribution of a
continuous r.v.. The normal curve is like:

The r.v. following a normal distribution is
called a normal r.v.
Finding Probability
1.
2.
3.
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Y: a normal r.v. with mean  and
standard deviation 
y
Finding z scores z 

Shade the required area under the
standard normal curve
Use Z-Table (p. 1170) to find the
answer
Example
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Suppose that the final scores of ST6304
students follow a normal distribution with  =
80 and  = 5. What is the probability that a
ST6304 student has final score 90 or above
(grade A)?
Between 75 and 90 (grade B)?
Below 75 (Fail)?
Sampling Distribution
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A parameter is a numerical summary of a
population, which is a constant.
A statistic is a numerical summary of a
sample. Its value may differ for different
samples.
The sampling distribution of a statistic is
the distribution of possible values of the
statistic for repeated random samples of the
same size taken from a population.
Sampling Distribution of Sample
Mean
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Example: suppose the pdf of a r.v. X is as
follows:
x
0
1
3
f(x)
0.5
0.3
0.2
Its mean 0.9 and variance 21.29.
Sampling Distribution of Sample
Mean
All possible samples of n=2:
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Sampling Distribution of Sample
Mean
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Sampling Distribution of Sample
Mean
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Central Limit Theorem
 y   and  y   / n
When n is large,
the distribution of y is approximately normal.
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Central Limit Theorem
(uniform[0,1])
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Normal Approximation to Binomial
Distribution

The binomial distribution is approximately
normal when the sample size is large
enough:
np  5; n1  p   5
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Continuity correction
Others
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
Random sampling and Normality checking
are in Lab 2

Poisson Distribtion