Transcript 5-1 Random Variables and Probability Distributions

5-1 Random Variables and
Probability Distributions
The Binomial Distribution
Random Variables
Discrete – These variables take on a finite number of
values, or a countable number of values
Number of days absent
Number of students taking a course
Continuous – These variables can take on an infinite
number of values on a number line
Time it takes for students to drive home
Gallons of gas you buy each time you go to the gas
station
(Typically length, temperature, volume, time, etc)
Probability Distribution
A probability distribution is an
assignment of probabilities to the
specific values of a random variable,
or to a range of values of the random
variable.
Discrete: probability assigned to each
value of the random variable (and the
sum = 1)
What does this mean?
Lets look at a graph of a probability
distribution of the discrete model, and
see what it is:
30.00
25.00
20.00
Series1
15.00
10.00
5.00
0.00
1
2
3
4
5
6
7
8
9
10
How do we look at it?
30.00
25.00
20.00
Series1
15.00
10.00
5.00
0.00
1
2
3
4
5
6
7
8
9
10
It looks like
A histogram, where the height is the
relative probability (i.e. a RELATIVE
FREQUENCY) and the bin is the
particular number.
Does this look familiar??
So what is the probability of choosing a
7 or a 3?
P(7 or 3) =
Mean and Standard Deviation
Probability distributions have a mean and
standard deviation.
For discrete population probability
distributions, the mean and the standard
deviation are given by formulas…
Which letters will we use?  and s or μ and σ?
Population…. Anyone??
Mean and Standard Deviation
μ   xP(x)
σ
(x  μ) P(x)
2
Sometimes this is called
the expected value of a
distribution - it is an
AVERAGE value, or what
can be thought of as a
central point (cluster
point)
Where x is the value of the random variable,
P(x) is the probability of that variable and
The sum is taken for all the values of that
random variable.
Notice – we are now discussing mean and
standard deviation of something other than
= Risk – the likelihood that a random variable is
raw data…
different from the mean
Lets consider the ways you can
toss a coin four times
Assume the coin is balanced (i.e. H and
T equally likely)
Assume also there is no memory (the
coin doesn’t remember that the last
toss was heads).
There are sixteen outcomes, right? (1/2)4
Lets draw out ALL possible outcomes
To assign a discrete random
variable
Let x = heads, so
x=0
TTTT
x=1
HTTT, THTT, TTHT, TTTH
x=2
HHTT, HTHT, HTTH, THHT, THTH,
TTHH
x=3
HHHT, HHTH, HTHH, THHH
x=4
HHHH
So the probability model would
be…
# of heads
0
1
2
3
4
Probability
.0625
.25
.375
.25
.0625
Calculate the mean and SD of
the distribution
mean = 2
Standard Deviation = 1
Linear Functions of Random
Variables
Suppose I have a and b, which are
constants. A new function L = a + bx
(where x is a random variable) ALSO
has a mean, variance and standard
deviation.
Linear Functions of Random
Variables
Suppose I have a and b, which are
constants. A new function L = a + bx
(where x is a random variable) ALSO
has a mean, variance and standard
deviation.
μL  a  bμ
2
L
2
σ b σ
σL  b σ
2
Combining independent
random variables
To make a linear combination of two
independent random variables x1 and
x2 ,
W = ax1 + bx2
and
Combining independent
random variables
To make a linear combination of two
independent random variables x1 and
x2 ,
W = ax1 + bx2
and
μW  aμ1  bμ2
2
W
2
2
1
2
σ  a σ b σ
2
2
σW  a2σ12  b2σ22
What?
All this is a way to look what happens
when you transform data (for instance,
if I take all the data and multiply by 2
then add 10, for rescaling purposes).
Application
The manager of a computer company quickly
shipped 2 computers to a client on the
same day as the order. Unfortunately, the
two computers were accidentally chosen
from a stockroom with an inventory of 15
computers, 4 of which were refurbished.
If one of the computer is refurbished it will be
sent back at your expense ($100). If both
are refurbished, the client will cancel the
order this month and you will lose $1000.
What is the expected value and standard
deviation of your loss?
Sources
http://www.mathworks.com/access/helpdesk/help/toolbox/stats
/index.html?/access/helpdesk/help/toolbox/stats/f4218.htm