6.3. Expected Value and Variance

6.3. Expected Value and Variance:

(I): Discrete Random Variable:

(a) Expected Value:

Example:

X: the random variable representing the point of throwing a fair dice. Then,

Intuitively, the average point of throwing a fair dice is

The expected value of the random variable X is just the average,

Example A (continue):

In the gambling example,

Intuitively (or ideally), the tokens gained or lost for 6 bets should be

3 / -4 / -4 / -4 / -4 / 0

That is, the relative frequency for the data 3, -4, and 0 are and , respectively. The average token number is

Since the relative frequency plays the role of the probability distribution, while the the values of the random variable plays the role of the data values, it is sensible to define the expected value of the random variable Xby

Therefore, on the average, the gambler will lose for every bet.

Similarly, the expected value of the random variable Y is

Formula for the expected value of a discrete random variable:

Let be all the possible values of the discrete random variable X and is the probability distribution. Then, the expected value of the discrete random variable X is

Note: As the grouped data available are with frequencies , respectively, i.e., with relative frequencies

the average of the data is

where

(b) Variance:

Example:

Suppose we want to measure the variation of the random variable X in the dice example. Then, the square distance between the values of X and its mean E(X)=3.5 can be used, i.e., can be used. The average square distance is

Intuitively, large average square distance implies the values of X scatter widely.

The variance of the random variable X is just the average square distance (the expected value of the square distance). The variance for the dice example is

Example A (continue):

Intuitively (or ideally), the tokens gained or lost for 6 bets should be

3 / -4 / -4 / -4 / -4 / 0

That is, the relative frequency for the data 3, -4, and 0 are and , respectively. The average of the data is

. Therefore, the variance for the data, considered as the population, is

Similarly, the variance of the random variable Y is

Formula for the variance of a discrete random variable:

Let be all the possible values of the discrete random variable X and is the probability distribution. Let be the expected value of X. Then, the variance of the discrete random variable X is

Note: As the grouped data available are with frequencies , respectively, i.e., with relative frequencies

the variance of the data, considered as the population, is

where

Example 1:

The probability distribution function for a discrete random variable X is

where k is some constant. Please find

(a) k. (b) (c) and

[solution:]

(a)

(b)

(c)

and

(II): Continuous Random Variable:

(a) Expected Value:

Example B (continue):

Z: the random variable representing the delay flight time taking values in [0,1].

Then, the probability density function for Z is

Intuitively, since there is equal chance for any delay time in [0,1], 0.5 hour seems to be a sensible estimate of the average delay time.

The expected value of the random variable Z is just the average delay time.

Formula for the expected value of a continuous random variable:

Let the continuous random variable X taking values in [a,b] and is the probability density function. Then, the expected value of the continuous random variable X is

Example B (continue):

In the flight time example, suppose the probability density function for Z is

Then, the expected value of the random variable Z is

Therefore, on the average, the flight time is hour.

(b) Variance:

Example B (continue):

Suppose we want to measure the variation of the random variable Z in the flight time example. Suppose is the probability density function for Z. Then, the square distance between the values of Z and its mean can be used, i.e., can be used. The average square distance is

The variance of the random variable Z is just the average square distance (the expected value of the square distance). The variance for the flight time example is

Formula for the variance of a continuous random variable:

Let the continuous random variable X taking values in [a,b] and is the probability distribution. Let be the expected value of X. Then, the variance of the continuous random variable X is

Example B (continue)

In the flight time example, suppose is the probability density function for Z.

Then, the variance of the random variable Z is

Example 2:

The probability density function for a continuous random variable X is

where a, b are some constants. Please find

(a)a, b if (b) .

[solution:]

(a)

and

Solve for the two equations, we have

(b)

Thus,

Example 3:

The probability density function for a continuous random variable X is

Please find (a) (b) (c) and

[solution:]

(a)

(b)

(c)

Since

Example 4:

The probability distribution functions (discrete random variable) or probability density functions(continuous random variable) for a random variable X are

(a)

(b)

(c)

Find .

[solution:]

(a)

(b)

(c)

Online Exercise:

Exercise 6.3.1

Exercise 6.3.2