Econ 215 Chapter 4

# Econ 215 Chapter 4 - Chapter 4 Equilibrium Theory of Bond...

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Chapter 4 Equilibrium Theory of Bond Markets

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Random Variables A random variable is a variable whose value is determined by chance. A random variable X is specified by a list of its possible values and a list of the corresponding probabilities: X 1 X 2 X 3 X n p 1 p 2 p 3 p n
The list of probabilities is called a probability distribution. Notation : The probability distribution for the random variable X is denoted by One interpretation of probability is that it represents relative frequency. ( ) : ( ) p x P X x = =

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A probability distribution is mathematically specified by the following two conditions: 0 ( ) 1 ( ) 1 x p x p x =
Mathematical Expectations The expectation of the random variable X is defined by Alternative expressions for expectation include mean and expected value . : ( ) e x X xp x =

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The variance of the random variable X is defined by The standard deviation of the random variable X is defined by In other words, it is the square root of the variance. 2 2 : ( ) ( ) e x x X p x σ = - 2 : σ σ =
Interpretation The expectation is a measure of the center of the probability distribution of X. The variance and standard deviation are measures of the variation of the probability distribution of X.

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Example Compute the mean and standard deviation for the following random variable: X 1 2 3 4 P .2 .3 .4 .1
4 3 2 1 4 3 2 1 0 C1 Frequency Histogram

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Risk Measurement We now enter a world in which future prices and interest rates are random variables.
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