Transcript Rules_for_Means_Variance

Rules for Means and Variances
Rules for Means:
Rule 1: If X is a random variable and a and b are
constants, then
a bX  a  bX
If we add a constant a to every value of a random
variable, then its mean will have a added, as well.
If we multiply every value of a random variable
by a constant, then its mean will be multiplied
by that constant, also.
Let’s try an example. Let X be the random variable shown:
X
0
1
2
3
4
P(X)
.1
.3
.2
.3
.1
We found in an earlier exercise that X=2. Suppose we
now find 3+1.5X.
3+1.5X
3
4.5
6
7.5
9
P(3+1.5X)
.1
.3
.2
.3
.1
By both direct calculation and applying the rule for
means, we find that X=6. ( (2 1.5)  3  6 )
Check this for yourself.
Rules for Means:
Rule 2: If X and Y are random variables, then
X Y  X  Y
When we find the sum of two random variables,
the mean is the sum of the individual random
variables.
Rules for Means:
Rule 2a: If X and Y are random variables, then
X Y  X  Y
When we find the difference of two random
variables, the mean is the difference of the
individual random variables.
Now for some examples, suppose the mean of X is 5 and the
mean of Y is 3.
To find X+Y we add X and Y .
To find X-Y we subtract X and Y .
That’s not so hard!
So, X+Y =5+3=8.
So, X-Y =5-3=2.
Rules for Variances:
Rule 1: If X is a random variable and a and b are
constants, then

2
a bX
b
2
2
X
If we add a constant a to every value of a random
variable, then its variance is unchanged.
If we multiply every value of a random variable
by a constant, then its variance will be
multiplied by the square of that constant.
Let’s try an example.
X
0
1
2
3
4
P(X)
.1
.3
.2
.3
.1
2

We found in an earlier exercise that X  1.4 .
2

we now find 31.5X .
3+1.5X
P(3+1.5X)
3
.1
4.5
.3
Suppose
6
7.5
9
.2
.3
.1
By both direct calculation and applying the rule for
2
2
variances, we find that 31.5X  3.15 (1.5 1.4  3.15).
Check this for yourself.
Rules for Variances:
Rule 2: If X and Y are independent random variables,
then

2
X Y
 
2
X
2
Y
When we find the sum of two random variables,
the variance is the sum of the individual variances
of the random variables, provided they are
independent.
Rules for Variances:
Rule 2a: If X and Y are independent random variables,
then

2
X Y
 
2
X
2
Y
When we find the difference of two random
variables, the variance is the sum of the
individual variances of the random variables,
provided they are independent.
Note that we still add, not find the difference!!
Rules for Variances:
Rule 3: If X and Y are random variables with
correlation , then

2
XY
     2  X Y
2
X
2
Y
When we find the sum of two random variables,
the variance is the sum of the individual variances
of the random variables plus the additional term
2XY.
Rules for Variances:
Rule 3a: If X and Y are random variables with
correlation , then

2
XY
     2  X Y
2
X
2
Y
When we find the difference of two random variables,
the variance is the sum of the individual variances of
the random variables plus the additional term 2XY.
Note that we did not subtract, but added the variances!!
Now for some examples, suppose X and Y are independent,
and the standard deviation of X is 2.4 and the standard
deviation of Y is 1.7.
To find 
So,

2
XY
To find 
2
X Y
we add  and  .
2
X
2
Y
 2.4  1.7  8.65.
2
2
X Y
2
we add  and .
2
X
2
Y
2
2
2


2.4

1.7
 8.65.
So, XY
Notice that they are the same, whether we add or subtract the
random variables.
Now for some examples: suppose X and Y are random
variables with a correlation of .7, and the standard
deviation of X is 2.4 and the standard deviation of Y is 1.7.
2
2
2



To find X Y we add X and Y and 2  X Y .
2
2
2


2.4

1.7
 2(.7)(2.4)(1.7)  5.712.
So, XY
2
2
2



To find X Y we add X and Y and 2  X Y .
2
2
2


2.4

1.7
 2(.7)(2.4)(1.7)  5.712.
So, XY
Notice that they are the same, whether we add or subtract the
random variables.
These rules for combinations of random variables are
extremely important. They will not be provided on
your formula sheet!
THE END