Properties of Random Variables

1
Properties of Random Variables

• PDF completely summarizes the random variable,
  but does not help to test our economic question.
  For example, whether stock prices are expected to
  rise or fall in a given day?
• Expected value (or population mean value) The
  expected value of a discrete random variable X
  is

2
Properties of Expected Value
3
Properties of Expected Value

• The law of large numbers Suppose one repeatedly
  observes different realized values of a random
  variable and calculates the mean of the realized
  values. The mean will tend to be close to the
  expected value the more times one observes the
  random variable, the closer the mean will tend to
  be.
• The expected value of a random variable (say,
  stock price) does not tell us how much it will go
  up or down. The variance provides a measure of
  how far the random variable is likely to be away
  from its mean.

4
Properties of Random Variables

• For a discrete random variable, the variance (?2
  E(X - ?)2 is calculated by
• Since the variance is the average value of the
  squared distance between Xi and ?, it does not
  have an easy interpretation.
• The standard deviation is a very useful measure.
  The standard deviation ? of a random variable is
  equal to the square root of the variance of the
  random variable.

5
Application of Standard Deviation

• May June 2001 Dell Computer Stock Price and
  Yahoo Stock price. Average Yahoo stock price
  19.07 and Average Dell Stock Price 25.11.
• Var(Yahoo stock price change) 1.324
• Stdv (Yahoo stock price change) 1.15
• Var(Dell stock price change) 0.524
• Stdv (Dell stock price change) 0.724
• Note The standard deviation is bigger for Yahoo
  than for Dell. Which implies that the daily stock
  price changes are more variable (farther away
  from the mean) for Yahoo than for Dell. Ceteris
  Paribus, Which stock is riskier?

6
Properties of Variance

• 1. Var(constant) 0
• 2. If X and Y are two independent random
  variables, then
• Var(X Y) Var(X) Var (Y) and
• Var(X - Y) Var(X) Var (Y)
• 3. If b is a constant then Var(bX) Var(X)
• 4. If a is a constant then Var(aX) a2Var(X)
• 5. If a and b are constants then Var(aXb)
  a2Var(X)
• 6. If X and Y are two independent random
  variables and a and b are constants then
  Var(aXbY) a2Var(X) b2Var(Y)

7
Covariance

• Covariance For two discrete random variables X
  and Y with E(X) ?x and E(Y) ?y, the
  covariance between X and Y is defined as Cov(XY)
  ?xy E(X - ?x) E(Y - ?y) E(XY) - ?x ?y.
• To computer the covariance, we use the following
  formula

8
Covariance

• In general, the covariance between two random
  variables can be positive or negative. If two
  random variables move in the same direction, then
  the covariance will be positive, if they move in
  the opposite direction the covariance will be
  negative.
• Properties
• 1.If X and Y are independent random variables,
  their covariance is zero. Since E(XY) E(X)E(Y)
• 2. Cov(XX) Var(X)
• 3. Cov(YY) Var(Y)

9
Correlation Coefficient

• The covariance tells the sign but not the
  magnitude about how strongly the variables are
  positively or negatively related. The correlation
  coefficient provides such measure of how strongly
  the variables are related to each other.
• For two random variables X and Y with E(X) ?x
  and E(Y) ?y, the correlation coefficient is
  defined as

10
Correlation Coefficient

• 1. Like the covariance, the correlation
  coefficient can be positive or negative same
  sign as the covariance.
• 2. The correlation coefficient always lies
  between 1 and 1. 1 perfectly negatively
  correlated and 1 perfectly positively
  correlated.
• 3. Variances of correlated variables
• Var(X Y) Var(X) Var(Y) 2Cov(X,Y)
• Var(X - Y) Var(X) Var(Y) 2Cov(X,Y)