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Discrete Random Variables And Probability Distributions



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Discrete Random Variables And Probability Distributions

  1. 1. 1.4 Discrete Random Variables and Probability Distributions<br />
  2. 2. Random Variables<br />Definition: A random variable X on a sample space S is a rule that assigns a numerical value to each outcome of S or in other words a function from S into the set R of real numbers.<br />X : S  R<br />x : value of random variable X<br />RX : The set of numbers assigned by random variable X, i.e. range space.<br />
  3. 3. Random Variables <br />Classifications of Random Variables According to the number of values which they can assume, i.e. number of elements in Rx.<br />Discrete Random Variables: Random variables which can take on only a finite number, or a countable infinity of values, i.e. Rx is finite or countable infinity.<br />Continuous Random Variables: When the range space Rx is a continuum of numbers. For example an interval or the union of the intervals.<br />
  4. 4. Random Variables <br />Example: Consider the experiment consisting of 4 tosses of a coin then sample space is<br />S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTTH, TTHH, THTH, HTHT, THHT, TTTH, TTHT, THTT, HTTT, TTTT}<br />Let X assign to each (sample) point in S the total number of heads that occurs. Then X is a random variable with range space <br />RX = {0, 1, 2, 3, 4}<br /> Since range space is finite, X is a discrete random variable <br />
  5. 5. Random Variables <br />Example: A point P is chosen at random in a circle C with radius r. Let X be the distance of the point from the center of the circle. Then X is a (continuous) random variables with RX = [0, r]<br />P<br />C<br />r<br />X<br />O<br />
  6. 6. Probability Distributions<br />If X is discrete random variable, the function given by <br />f(x) = P[X = x]<br /> for each x within the range of X is called the probability mass function (pmf) of X.<br />To express the probability mass function, we give a table that exhibits the correspondence between the values of random variable and the associated probabilities<br />
  7. 7. Probability Distributions <br />Ex: In the experiment consisting of four tosses of a coin, assume that all 16 outcomes are equally likely then probability mass function for the total number of heads is <br />
  8. 8. Probability Distributions <br />A function can serve as the probability mass function of a discrete random variable X if and only if its value, f(x), satisfy the conditions<br />1. f(x)  0 for all value of x. <br /> 2.<br />Example: Check whether the following can define <br />probability distributions<br />
  9. 9. Probability Distributions <br />Ans: (a) Yes, (b) No, (c) Yes, (d) No<br />
  10. 10. Distribution Function<br />If X is a discrete random variable, the function given by<br />where f(t) is the value of the probability mass function<br />of X at t, is called the distribution function or the<br />cumulative distribution function (cdf) of X.<br />
  11. 11. Example<br />Distribution function of the total number of heads obtained in four tosses of a balanced coin<br />We know that f(0) = 1/16, f(1) = 4/16, f(2) = 6/16, f(3) = 4/16, f(4) = 1/16. It follows that<br />F(0) = f(0) = 1/16 <br />F(1) = f(0) + f(1) = 5/16<br />F(2) = f(0) + f(1) + f(2) =11/16<br />F(3) = f(0) + f(1) + f(2) + f(3) = 15/16<br />F(4) = f(0) + f(1) + f(2) + f(3) + f (4) = 1<br />
  12. 12. The distribution function is given by<br />
  13. 13. F(x)<br />.<br />.<br />1<br />15/16<br />.<br />11/16<br />.<br />5/16<br />.<br />1/16<br />x<br />2<br />3<br />4<br />0<br />1<br />Graph of the Distribution function<br />
  14. 14. Distribution Function <br /> The distribution function is defined not only for the values taken on by the given random variable, but for all real number.<br /> We can write F(1.7) = 5/16 and F(100) = 1, although the probability 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.<br />
  15. 15. Distribution Function <br /><ul><li>The values F(x) of the distribution function of a discrete random variable X satisfy the conditions</li></ul>F(-) = 0 and F() = 1; that is, it ranges from 0 to 1.<br />If a &lt; b, then F(a)  F(b) for any real numbers a and b. Hence it is non-decreasing.<br /><ul><li>If the range of a random variable X consists of the values x1 < x2 < x3 < … < xn, then f(x1) = F(x1) and</li></ul> f (xi) = F(xi) - F(xi-1) for i = 2, 3, …, n. <br /> That is, f (xi) is the size of the jump in the graph, <br />