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This section presents various fundamental concepts related to probability and random variables that will be required to support subsequent discussions The discussion will focus on continuous random variables

Let w represent a real, scalar random variable In the following, the notation P {w < W } denotes the probability that the random variable w is less than the real number W The distribution function of the random variable w is de ned as the function (48) Fw (W ) = P {w W } The function Fw (W ) is monotonically increasing in W , is continuous from the right with Fw ( ) = 0 and Fw ( ) = 1 The derivative of the distribution function is the density function of the random variable w and will

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Many traders and investors have said that if they were forced to choose only one indicator, it would probably be moving averages Even some investors who almost never look at indicators pay attention to moving averages Why Because this indicator is powerful yet easy to use and simple to understand on a chart Basically, moving averages show the value of a security s price over the duration of a period of time, such as the last 20, 50, 100, or 200 days By overlaying the moving average on top of the stock or market, you get a good feel for the direction of the market Traders who use trend-following strategies are drawn to moving averages Also, when moving averages are displayed on a chart, it s much easier to see trend changes For example, if the market falls below one of the moving averages, that could be a signal the current trend is ending Perhaps one of the best reasons to use moving averages is to help keep your emotions out of the trading process For example, if you created a rule to buy when the market crossed above the 50day moving average (and stayed there), and a rule to sell when the market crossed back below, you d be surprised at what can be achieved Sometimes it s hard to believe that a system so easy could be so profitable Nevertheless, many traders tweak the defaults, using a variety of different time periods

dFw (W ) dW The density function has the interpretation that pw (W ) represents the probability that the random variable w assumes a value in the di erential range (W, W + dW ] Because the density is the derivative of a monotone function, pw (W ) 0 for any W Based on the above de nitions, for a scalar random variable, pw (W ) = P {W1 < w W2 } = Fw (W2 ) Fw (W1 ) =

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E g(w) =

(49)

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