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      CONTINUOUS RANDOM VARIABLES

      We can also define the mean, median, and mode for a continuous random variable. To find the mean of a continuous random variable, we simply integrate the product of the variable and its probability density function (PDF). In the limit, this is equivalent to our approach to calculating the mean of a discrete random variable. For a continuous random variable, X, with a PDF, f(x), the mean, μ, is then:

      (3.26)

      The median of a continuous random variable is defined exactly as it is for a discrete random variable, such that there is a 50 percent probability that values are less than or equal to, or greater than or equal to, the median. If we define the median as m, then:

      (3.27)

      Alternatively, we can define the median in terms of the cumulative distribution function. Given the cumulative distribution function, F(x), and the median, m, we have:

      (3.28)

      The mode of a continuous random variable corresponds to the maximum of the density function. As before, the mode need not be unique.

      On January 15, 2005, the Huygens space probe landed on the surface of Titan, the largest moon of Saturn. This was the culmination of a seven-year-long mission. During its descent and for over an hour after touching down on the surface, Huygens sent back detailed images, scientific readings, and even sounds from a strange world. There are liquid oceans on Titan, the landing site was littered with “rocks” composed of water ice, and weather on the moon includes methane rain. The Huygens probe was named after Christiaan Huygens, a Dutch polymath who first discovered Titan in 1655. In addition to astronomy and physics, Huygens had more prosaic interests, including probability theory. Originally published in Latin in 1657, De Ratiociniis in Ludo Aleae, or The Value of All Chances in Games of Fortune, was one of the first texts to formally explore one of the most important concepts in probability theory, namely expectations.

      Like many of his contemporaries, Huygens was interested in games of chance. As he described it, if a game has a 50 percent probability of paying $3 and a 50 percent probability of paying $7, then this is, in a way, equivalent to having $5 with certainty. This is because we expect, on average, to win $5 in this game:

      (3.29)

      As one can already see, the concepts of expectations and averages are very closely linked. In the current example, if we play the game only once, there is no chance of winning exactly $5; we can win only $3 or $7. Still, even if we play the game only once, we say that the expected value of the game is $5. That we are talking about the mean of all the potential payouts is understood.

      We can express the concept of expectation more formally using the expectations operator. We could state that the random variable, X, has an expected value of $5 as follows:

      (3.30)

      where E[ ·] is the expectation operator.1

      In this example, the mean and the expected value have the same numeric value, $5. The same is true for discrete and continuous random variables. The expected value of a random variable is equal to the mean of the random variable.

      While the value of the mean and the expected value may be the same in many situations, the two concepts are not exactly the same. In many situations in finance and risk management the terms can be used interchangeably. The difference is often subtle.

      As the name suggests, expectations are often thought of as being forward-looking. Pretend we have a financial asset for which the mean annual return is equal to 15 percent. This is not an estimate; in this case, we know that the mean is 15 percent. We say that the expected value of the return next year is 15 percent. We expect the return to be 15 percent, because the probability-weighted mean of all the possible outcomes is 15 percent.

      Now pretend that we don't actually know what the mean return of the asset is, but we have

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