Discrete R.V.
1 Random Variable
Definition 1 A random variable (rv) is a numerical variable whose value depends on the outcome of an experiment.
A random variable must be a number; it cannot be a letter, say. More precisely, a random variable is a “real-valued function for which the domain is a sample space”.
Example 1 A coin is tossed twice and the sequence of \(H\)’s and \(T\)’s is observed. Let \(Y\) be the number of \(H\)’s which come up. Show that \(Y\) is a random variable. The experiment here has 4 possible outcomes: \(TT\), \(TH\), \(HT\), \(HH\).
\(Y = 0\) if the outcome is \(TT\)
\(Y = 1\) if the outcome is \(TH\) or \(HT\)
\(Y = 2\) if the outcome is \(HH\)
1.1 Probability Distribution
The probability that a discrete random variable \(Y\) takes on a particular value \(y\) is the sum of the probabilities of all sample points in the sample space \(S\) that are associated with \(y\).
We write this probability \(P(Y = y)\).
The probability distribution of a discrete random variable \(Y\) is any information which provides \(P(Y = y)\) for each possible value \(y\) of \(Y\). This information may take the form of a list, table function (formula) or graph.
1.2 Probability Mass Function
It is conventional to denote rv’s by upper case letters (e.g., \(Y\), \(X\), \(U\)) and possible values of those rv’s by the corrsponding lower case letters (e.g., \(y\), \(x\), \(u\)).
\(P(Y = y)\) is called the probability mass function (pmf) of \(Y\) and is often written \(p(y)\) or \(p_Y(y)\).
1.2.1 Two Properties of Discrete PMF’s
- \(0 \leq p(y) \leq 1\) for all \(y\)
- \(\sum_y p(y) = 1\)
Example 2 A coin is repeatedly tossed until the first head comes up. Let \(Y\) be the number of tosses. Derive the pmf of \(Y\), and check that it satisfies the two properties of discrete pmf’s.
Then \(Y\) has pmf, \[ p(y) = \left( \frac{1}{2} \right)^y \]
We should also observe that Property 1 is satisfied, since 1/2, 1/4, 1/8, … are all between o and 1. Also,
\[ \sum_y p(y) = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1 \]
Thus Property 2 is also satisfied.
\(Y\) is a discrete rv in this example because \(\{1,2,3,\ldots\}\) is a countably infinite set (its elements can be listed). A pmf uniquely defines a rv or pr dsn. Thus a rv can t have 2 or more different pmf’s. Note that not all functions are valid pmf’s.