Transcript Example

PROBABILITY AND
STATISTICS
WEEK 4
Onur Doğan
Random Variable
Random Variable. Let S be the sample space for an experiment. A
real-valued function that is defined on S is called a random
variable.
Onur Doğan
Random Variable
1.Discrete Random Variable: Has a finite (or
countably infinite) range.
•Tossing a coin: X= 0 for head and X= 1 for tail
2.Continuous Random Variable: Has an interval
of real numbers for its infinite range.
•The life length of a light bulb: X ≥ 0
Onur Doğan
Reminder !
1.
x
is the mean of the sample
2.
s2 and s are the variance and standard deviation of the sample
3.
x , s2, and s are called sample statistics
4.
m (lowercase Greek letter “mu”) is the mean of the population
5.
s2 (“sigma squared”) is the variance of the population
6.
s (lowercase Greek letter “sigma”) is the standard deviation of the population
7.
m, s2, and s are called population parameters. (A parameter is a constant. m, s2, and s
are typically unknown values.)
Discrete Random Variables
Let 4 coins tossed, and let X be the number of heads
that are obtained. Let us find the distributions of that
experiment.
Onur Doğan
Probability Distribution
Onur Doğan
Discrete Random Variables
Example
Three balls, a, b, c, are randomly distributed in three
boxes. Determine the distribution of the random
variable X ="the number of non-empty boxes".
Onur Doğan
Discrete Random Variables
Example
Consider a group of five potential blood donors;
“a, b, c, d, and e” of whom only a and b have type 0+
blood.
Five blood samples, one from each individual, will be
typed in random order until an 0+ individual is
identified. Let the rv Y=“the number of typings
necessary to identify an 0+ individual.”
Find the pmf.
Onur Doğan
The Cumulative Distribution Function
Onur Doğan
Example
Onur Doğan
The Expected Value of X
(Mean of a Discrete Random Variable)
Onur Doğan
The Expected Value of X
(Mean of a Discrete Random Variable)
• The mean, m, of a discrete random variable x is
found by multiplying each possible value of x by its
own probability and then adding all the products
together:
m =
 [ x.p ( x )]
Notes:

The mean is the average value of the random variable, what happens on average

The mean is not necessarily a value of the random variable
The Variance of X
Onur Doğan
Example
The grades of n = 50 students in a statistics class
are summarized as follows:
Grade (X)
Number of
Students
1
2
3
4
10
20
15
5
Find the pmf, mean, variance and sd.
Onur Doğan
Example
Variance and Standard Deviation of a Discrete
Distribution. Suppose that a random variable X can take
each of the five values −2, 0, 1, 3, and 4 with equal
probability.
Determine the variance and standard deviation of X.
Onur Doğan
A Shortcut Formula for V(X)
Onur Doğan
Example
Determine the mean, variance, and standard
deviation of casting a single die (X).
Onur Doğan
Example
Onur Doğan
Example
A shipment of 8 similar microcomputers to
contains 3 defective one. If a school makes a
random purchase of 2 of these computers, find
the probability distribution for the number of
defectives.
Onur Doğan
Example
Onur Doğan
Example
Example: The probability distribution for a random
variable x is given by the probability function:
8 x
P( x ) =
15
for
x = 3, 4, 5, 6, 7
Find the mean, variance, and standard deviation
Discrete Uniform Distribution
A discrete uniform random variable X has an equal
probability for each value in the range of X= [a, b],
a < b. Thus, the probability mass function of X is;
P(x)= 1/(b-a+1)
where
Onur Doğan
x=a,a+1,…,b
Example
• Casting a die…
Onur Doğan
Example
Suppose that product codes of 2, 3, or 4 letters
are equally likely.
• Determine the probability mass function of the
number of letters (X) in a product code.
• Calculate the mean and variance of X
Onur Doğan