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Continuous Random Variables Suppose we are interested in the probability that a given random variable will take on a value on the interval from a to b where a and b are constants with a b. First, we divide the interval from a to b into n equal subintervals of width x containing respectively the points x1, x2, … , xn. Suppose that the probability that the random variable will take on a value in subinterval containing xi is given by f(xi)x. Then the probability that the random variable will take on a value in the interval from a to b is given by
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Continuous Random Variables(cont’d) If f is an integrable function defined for all values of the random variable, the probability that the value of the random variables falls between a and b is defined by letting x 0 as Note: The value of f(x) does not give the probability that the corresponding random variable takes on the values x; in the continuous case, probabilities are given by integrals not by the values f(x).
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Continuous Random Variables(cont’d) f(x) P(a X b) a b x Figure: Probability as area under f
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Continuous Random Variables(cont’d) The probability that a random variable takes on value x, i.e. Thus, in the continuous case probabilities associated with individual points are always zero. Consequently,
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Continuous Random Variables(cont’d) The function f is called probability density function or simply probability density. Characteristics of the probability density function f : 1. for all x. 2. F(x) represents the probability that a random variable with probability density f(x) takes on a value less than or equal to x and the corresponding function F is called the cumulative distribution function or simply distribution function of the random variable X.
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Continuous Random Variables(cont’d) Thus, for any value x, F (x) = P(X x) is the area under the probability density function over the interval - to x. Mathematically, The probability that the random variable will take on a value on the interval from a to b is given by P(a X b) = F (b) - F (a)
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Continuous Random Variables(cont’d) According to the fundamental theorem of integral calculus it follows that wherever this derivative exists. F is non-decreasing function, F(-) = 0 and F() = 1. kth moment about the origin
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Continuous Random Variables(cont’d) Mean of a probability density: kth moment about the mean:
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Continuous Random Variables(cont’d) Variance of a probability density is referred to as the standard deviation
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