Continuous Random Variables

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Continuous Random Variables

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Continuous Random Variables

  1. 1. 1.7 Continuous Random Variables<br />
  2. 2. Continuous Random Variables<br />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. <br />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<br />
  3. 3. Continuous Random Variables(cont’d)<br />If f is an integrable function defined for all values of the<br />random variable, the probability that the value of the <br />random variables falls between a and b is defined by <br />letting x  0 as <br />Note: The value of f(x) does not give the probability that the <br />corresponding random variable takes on the values x; in the <br />continuous case, probabilities are given by integrals not by the <br />values f(x).<br />
  4. 4. Continuous Random Variables(cont’d)<br />f(x)<br />P(a  X  b)<br />a<br />b<br />x<br />Figure: Probability as area under f<br />
  5. 5. Continuous Random Variables(cont’d)<br />The probability that a random variable takes on value x, i.e.<br />Thus, in the continuous case probabilities associated with <br />individual points are always zero. Consequently,<br />
  6. 6. Continuous Random Variables(cont’d)<br />The function f is called probability density function or simply probability density. <br />Characteristics of the probability density function f :<br />1.<br />for all x.<br />2.<br />F(x) represents the probability that a random variable with <br />probability density f(x) takes on a value less than or equal to<br />x and the corresponding function F is called the cumulative<br />distribution function or simply distribution function of the <br />random variable X.<br />
  7. 7. Continuous Random Variables(cont’d)<br />Thus, for any value x, <br />F (x) = P(X  x)<br />is the area under the probability density function over the interval - to x. Mathematically,<br /> The probability that the random variable will take on a value <br /> on the interval from a to b is given by<br />P(a  X  b) = F (b) - F (a)<br />
  8. 8. Continuous Random Variables(cont’d)<br />According to the fundamental theorem of integral calculus it follows that <br />wherever this derivative exists.<br /> F is non-decreasing function, F(-) = 0 and F() = 1.<br />kth moment about the origin<br />
  9. 9. Continuous Random Variables(cont’d)<br />Mean of a probability density:<br />kth moment about the mean:<br />
  10. 10. Continuous Random Variables(cont’d)<br />Variance of a probability density<br /> is referred to as the standard deviation<br />

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