Probability and statistics II week 3
As in the discrete case, there are many problems in which it is of interest to know the probability that the value of a continuous random variable X is less than or equal to some real number x. Thus, let us make the following definition analogous to Definition 1.3. Definition 1.6 (Distribution Function or Cumulative Density Function) If X is a continuous random variable and the value of its probability density function at t is , the function given by for is called the Distribution Function, or the Cumulative Density Function, of X. Other properties which have to be obeyed by F (x) include: (i) (ii) F (x) is a monotone, non – decreasing function, i.e., Theorem 1.6 If and are the values of the probability density and the distribution function of X at x, then (i) for any real constants a and b with ; and (ii) , where the derivative exists. Example 1.13 Let X be a random variable of continuous type be defined by the p. d. f. Find the cumulative density function of X, F (x). Solution That is:
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- May 30, 2023
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- Prof kinyanjui
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- Week 3
