1. The document defines discrete random variables as random variables that can take on a finite or countable number of values. It provides an example of a discrete random variable being the number of heads from 4 coin tosses. 2. It introduces the probability mass function (PMF) as a function that gives the probability of a discrete random variable taking on a particular value. The PMF must be greater than or equal to 0 and sum to 1. 3. The cumulative distribution function (CDF) of a discrete random variable is defined as the sum of the PMF values up to that point. It ranges from 0 to 1 and increases monotonically.

1. 1.4 Discrete Random Variables and Probability Distributions

2. Random Variables 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. X : S  R x : value of random variable X RX : The set of numbers assigned by random variable X, i.e. range space.

3. Random Variables Classifications of Random Variables According to the number of values which they can assume, i.e. number of elements in Rx. 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. Continuous Random Variables: When the range space Rx is a continuum of numbers. For example an interval or the union of the intervals.

4. Random Variables Example: Consider the experiment consisting of 4 tosses of a coin then sample space is S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTTH, TTHH, THTH, HTHT, THHT, TTTH, TTHT, THTT, HTTT, TTTT} 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 RX = {0, 1, 2, 3, 4} Since range space is finite, X is a discrete random variable

5. Random Variables 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] P C r X O

6. Probability Distributions If X is discrete random variable, the function given by f(x) = P[X = x] for each x within the range of X is called the probability mass function (pmf) of X. To express the probability mass function, we give a table that exhibits the correspondence between the values of random variable and the associated probabilities

7. Probability Distributions 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

8. Probability Distributions 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 1. f(x)  0 for all value of x. 2. Example: Check whether the following can define probability distributions

9. Probability Distributions Ans: (a) Yes, (b) No, (c) Yes, (d) No

10. Distribution Function If X is a discrete random variable, the function given by where f(t) is the value of the probability mass function of X at t, is called the distribution function or the cumulative distribution function (cdf) of X.

11. Example Distribution function of the total number of heads obtained in four tosses of a balanced coin 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 F(0) = f(0) = 1/16 F(1) = f(0) + f(1) = 5/16 F(2) = f(0) + f(1) + f(2) =11/16 F(3) = f(0) + f(1) + f(2) + f(3) = 15/16 F(4) = f(0) + f(1) + f(2) + f(3) + f (4) = 1

12. The distribution function is given by

13. F(x) . . 1 15/16 . 11/16 . 5/16 . 1/16 x 2 3 4 0 1 Graph of the Distribution function

14. Distribution Function The distribution function is defined not only for the values taken on by the given random variable, but for all real number. 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.