Probability and Statistics II week 2

There are many problems in which it is of interest to know the probability that the value of a random variable is less than or equal to some real number x. Thus, let us write the probability that X takes on a value less than or equal to x as and refer to this function defined for all real numbers x as the distribution function, or the cumulative distribution function, of X Definition 1.3 (Distribution Function or Cumulative Distribution Function) If X is a discrete random variable, the function given by for each x within the range of X. where is the value of the probability distribution function of X at t, is called the Distribution Function, or the Cumulative Distribution Function, of X. Based on the postulates of probability and some of their immediate consequences, it follows that: Theorem 1.2. The values of the distribution function of a discrete random variable X satisfy the conditions (i) and ; (ii) If , then , for any real numbers a and b. If we are given the probability distribution of a discrete random variable, the corresponding distribution function is generally easy to find. Example 1.6 Find the distribution function of the total number of heads obtained in four tosses of a balanced coin. Solution Based on the probabilities obtained in example 3 above, we find that and , it follows that Hence, the distribution function is given by Observe that this distribution function is defined not only for the values taken on by the given random variable, but for all real numbers. For instance, we can write and , although the probabilities of getting “at most 1.7 heads” or “at most 100 heads” in four tosses of a balanced coin may not be of any real significance. Example 1.7 Find the distribution function of the random variable W of Example 2 above and plot its graph. Solution Based on the probabilities obtained in Example 2 above, we can write , and , so that Hence, the distribution function of W is given by