ecen303_lec7

# ecen303_lec7 - ECEN 303 Random Signals and Systems Lecture...

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ECEN 303: Random Signals and Systems Lecture 7: Continuous Random Variables Consider the random experiment of picking a real number between 0 and 1. The probability that the number picked is within any interval inside [0 , 1] is equal to the length of the interval. Let X be the number picked. Then X is a random variable which can take on any real numbers in [0 , 1]. The CDF of X can be calculated as F ( x ) = Pr( X x ) = 0 , x < 0 x, 0 x < 1 1 , x 1 Here the CDF of X is a continuous function, so Pr( X = x ) = 0 for any real value x . Therefore, unlike discrete random variables, the distribution of such X cannot be described by specifying the probability of individual outcomes, i.e., the PMF. A random variable with a continuous CDF is called a continuous random variable . The best way to describe the distribution of a continuous random variable is to use the so-called probability density function (PDF). 1 Probability density function Let X be a continuous random variable with CDF F ( x ). By deﬁnition, F ( x ) is a nondecreasing, continuous function. It is known that such F ( x ) can be written as F ( x ) = Z x -∞ f ( a ) da for some nonnegative function f ( a ). Any nonnegative function f ( a ) that satisﬁes Z x -∞ f ( a ) da = F ( x ) x R 1

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is called a probability density function (PDF) of X . If F ( x ) is diﬀerentiable, the PDF f ( x ) = d dx F ( x ) Example: If the CDF of X is given by F ( x ) = 0 , x < 0 x, 0 x < 1 1 , x 1 the PDF of X can be calculated as f ( x ) = d dx F ( x ) = 0 , x < 0 1 , 0 < x < 1 0 , x > 1 We do not particularly care about the values of f ( x ) at x = 0 , 1, because no matter what they are it will yield the same CDF. Note that the PDF of a continuous random variable must satisfy Z -∞ f ( x ) dx = F ( ) = 1 Just like CDF, all probability questions bout X can be answered in terms of the PDF. For example: Pr( X x ) = F ( x ) = R x -∞ f ( a ) da Pr( X > x ) = 1 - Pr( X x ) = 1 - F ( x ) = R -∞ f ( a ) da - R x -∞ f ( a ) da = R x f ( a ) da Pr( x 1 < X x 2 ) = Pr( X x 2 ) - Pr( X x 1 ) = R x 2 -∞ f ( a ) da - R x 1 -∞ f ( a ) da = R x 2 x 1 f ( a ) da Pr( x 1 X x 2 ) = Pr( x 1 < X x 2 ) + Pr( X = x 1 ) = Pr( x 1 < X x 2 ) = R x 2 x 1 f ( a ) da The operational meaning of PDF. Let Δ be a small positive number. The probability Pr( X ( x 0 - Δ / 2 ,x 0 + Δ / 2)) = Z x 0 / 2 x 0 - Δ / 2 f ( x ) dx Z x 0 / 2 x 0 - Δ / 2 f ( x 0 ) dx = f ( x 0 ) Z x 0 / 2 x 0 - Δ / 2 dx = f ( x 0 2
In words, the probability that random variable X is within a small interval centered around x 0 is proportional to the length of the interval, with the ratio being the PDF of X evaluated at x 0 . Hence we have

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ecen303_lec7 - ECEN 303 Random Signals and Systems Lecture...

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