for Fm = 12 to be true is not available, but can only be estimated in such a way that Fm is as close to 12 as possible.
Example 4.11 computing the median for a discrete random variable
Find the median m of the random variable X whose values are taken to be the outcomes of tossing a fair die. Note that the mean of this X has been computed to be 3.5 in Example
4.7. Solution:
Obviously, the pmf for X is: p1 = 16, p 2 = 16, …, p6 = 16.
By the definition of cdf, it is easy to see that the cdf F for X is: F1 = 16, F2 = 16 + 16 = 13, F3 = 16 +16 + 16 = 12, and so on.
Therefore, the median of X is 3, which is different from the mean of X already known to be 3.5.
10. Two other types of means: geometric and harmonic means
The above-mentioned mean of numerical data actually is the so-called arithmetic mean, because there are two other types of means, namely, geometric mean and harmonic
mean which have respective significant applications.
F. Variance
1. Concept of variance
Another property of a random variable other than the mean and median is its variance which describes the degree of scatter of the random variable values. The larger the
variance, the larger the scatter.
Conceptually, if the values of the random variable are all the same in the extreme case, then the variance of the random variable should be zero.
2. Definition of the variance of a random variable
Definition 4.7 ---
If X is a random variable with mean , then the variance of X, denoted by VarX, is defined by
VarX = E[X
2
].
The variance is computed after a normalization of the random variable values with respect to the mean.
3. An alternative formula for computing the variance
Proposition 4.2
The value of VarX may be computed alternatively by
VarX = E[X
2
E [X]
2
. Proof:
VarX = E[X
2
] =
x
x
2
p x
by the definition of mean =
x
x
2
2 +
2
px =
x
x
2
p x
2
x
xp x +
2
x
p x
by Corollary 4.1 = E[X
2
] 2
[X] +
2
by the definition of mean and
x
p x = 1 coming from Axiom 2
= E[X
2
] 2
2
+
2
by the definition of mean = E[X
2
]
2
.
Comments: In words, the above proposition says that the variance of a random variable is
equal to the expected value of X
2
minus the square of its expected value. Use of this proposition is often the easiest way to compute VarX.
Example 4.12
Compute VarX if X represents the outcome of rolling a fair die. Solution:
By Proposition 4.1, E[X
2
] = 1
2
16 + 2
2
16 + ... + 6
2
16 = 916.
Also, we know from the result of Example 4.7 that E[X] = 3.5 = 72.
By Proposition 4.2, VarX = E[X
2
E [X]
2 2
= 3512.
Corollary 4.2
If a and b are constants, then VaraX + b = a
2
VarX. Proof:
By Corollary 4.1, we have E[aX + b] = aE[X] + b = + b. Accordingly, we have
VaraX + b = E[aX + b E[aX + b]
2
] by the definition of variance
= E[aX + b
b
2
] = E[a
2
X
2
]
= a
2
E [X
2
] by Corollary 4.1
= a
2
VarX. by the definition of variance
4. Definition of standard deviation
Definition 4.8
The square root of VarX, Var
X , is called the standard deviation of X, and is denoted as SDX, i.e.,
SDX = Var
X .
G. The Bernoulli and Binomial Random Variables
