Download Statistical Experiment - Milan C-2

Statistical Experiment
A statistical experiment or
observation is any process by
which an measurements are
obtained
Examples of Statistical
Experiments
• Counting the number of books in the
College Library
• Counting the number of mistakes on a
page of text
• Measuring the amount of rainfall in your
state during the month of June
Random Variable
a quantitative variable that
assumes a value determined by
chance
Discrete Random Variable
A discrete random variable is a
quantitative random variable that can take
on only a finite number of values or a
countable number of values.
Example: the number of books in the
College Library
Continuous Random Variable
A continuous random variable is a
quantitative random variable that can
take on any of the countless number of
values in a line interval.
Example: the amount of rainfall in your
state during the month of June
Probability Distribution
an assignment of probabilities to
the specific values of the random
variable or to a range of values of
the random variable
Probability Distribution of a
Discrete Random Variable
• A probability is assigned to each value of
the random variable.
• The sum of these probabilities must be 1.
Probability distribution for the
rolling of an ordinary die
x
1
2
3
P(x)
1
6
1
6
1
6
4
1
6
5
1
6
6
1
6
Features of a Probability
Distribution
x
P(x)
1
1
6
2
1
6
3
1
6
4
1
6
5
1
6
6
Probabilities
must be between
zero and one
(inclusive)
1
6
6
 1
6
 P(x) =1
Probability Histogram
P(x)
1
6
|
|
1
|
2
|
3
|
4
|
5
|
6
Mean and standard deviation
of a discrete probability
distribution
Mean =  = expectation or expected
value,
the long-run average
Formula:
 =  x P(x)
Standard Deviation

2
(
x


)
 P( x )

Finding the mean:
x
P(x)
x P(x)
0
.3
0
1
.3
.3
2
.2
.4
3
.1
.3
4
.1
.4
1.4
 =  x P(x) =
1.4
Finding the standard deviation
x –
( x – ) 2
( x – ) 2 P(x)
x
P(x)
0
.3
– 1.4
1.96
.588
1
.3
– 0.4
0.16
.048
2
.2
.6
0.36
.072
3
.1
1.6
2.56
.256
4
.1
2.6
6.76
.676
1.64
Standard Deviation

 ( x   )  P ( x )  1.64 
2
1.28
Linear Functions of a Random
Variable
If a and b are any constants and x is a
random variable, then the new random
variable
L = a + bx
is called a linear function of a random
variable.
If x is a random variable with
mean  and standard deviation
, and L = a + bx then:
• Mean of L =  L = a + b 
• Variance of L = L 2 = b2  2
• Standard deviation of L =  L= the square
root of b2  2 = b 
If x is a random variable with
mean = 12 and standard
deviation = 3 and L = 2 + 5x
• Find the mean of L.
• Find the variance of
L.
• Find the standard
deviation of L.
• L = 2 + 5

• Variance of L = b2  2 =
25(9) = 225
• Standard deviation of L
= square root of 225 =

Independent Random
Variables
Two random variables x1 and x2
are independent if any event
involving x1 by itself is
independent of any event
involving x2 by itself.
If x1 and x2 are a random variables
with means  and and variances
 and  If W = ax1 + bx2 then:
• Mean of W =  W = a  + b 
• Variance of W = W 2 = a2  12 + b2  2
• Standard deviation of W =  W= the
square root of a2  12 + b2  2
Given x1, a random variable with
1 = 12 and  1 = 3 and x2 is a
random variable with  2 = 8 and  2
= 2 and W = 2x1 + 5x2.
• Find the mean of W.
• Find the variance of
W.
• Find the standard
deviation of W.
• Mean of W =
2(12) + 5(8) = 64
• Variance of W =
4(9) + 25(4) = 136
• Standard deviation
of W= square root
of 136  11.66