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### Random Variables and Distributions

Lecture 5: Stat 700

Basics of Random Variables and Probability Distributions

- Consider again the experiment of tossing three fair coins simultaneously. The sample space and corresponding probabilities are given by:
- S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
- Probabilities ={1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8}
- In a practical setting, “H” may represent “success” of a medical operation; “T” represents “failure.”
- In this situation we may not really be interested in the elementary outcomes of S, but rather our interest might be on the total number of “H” that occurred in the outcome.

Notion of a Random Variable

- When interest is on some numerical characteristic of the outcomes of the experiment, then we are led into consideration of random variables.
- Definition: A random variable, denoted by X, Y, etc., is a function or procedure that assigns a unique numerical value to each of the outcomes of the experiment. The set of possible distinct values of the variable is called its Range.

A Simple Example

- In the coin-tossing experiment described earlier, if we let X denote the random variable counting the number of “H” in the outcome, then the values of X associated with each of the 8 possible outcomes are:
- X(HHH) = 3, X(HHT) = 2, X(HTH) = 2, X(THH) = 2, X(HTT) = 1, X(THT) = 1, X(TTH) = 1, X(TTT) = 0
- Therefore, Range of X = {0, 1, 2, 3}.
- One of the advantages of dealing with random variables instead of the elementary events of the experiment is that we will be dealing with numbers, which we could add, multiply, etc., instead of crude outcomes that we could not do arithmetic operations.

Types of Random Variables

- Random variables can be classified into either discrete or continuous variables.
- Discrete variables are those whose range has a finite number or at most a countable number of values; while
- Continuous variables are those whose range contains an interval of the real line, hence has an uncountable number of values.
- The variable X in the example is clearly a discrete random variable.

Probability Distribution Function

- Definition: Given a discrete random variable X whose range is R = {x1, x2, x3, …}, its probability function, denoted by p(x), is a table, a graph, or a mathematical formula which provides the probabilities for each of the possible values in its range. In formal notation,

Determining the Probability Function

- For a discrete random variable X, the value of p(xj) is obtained by summing up the probabilities of all the elementary events whose X-value is xj.
- A simple illustration makes this immediately transparent.
- For the variable X which counts the number of “H” that occur in the toss of three fair coins, we obtain the probability function as follows:

Probability Function for X in Example

- The range of X is R = {0,1, 2, 3}. We have:
- p(0) = P(X = 0) = P(TTT) = 1/8 = .125
- p(1) = P(X = 1) = P(HTT) + P(THT) + P(TTH) = 1/8 + 1/8 + 1/8 = 3/8 = .375
- p(2) = P(X = 2) = P(HHT) + P(HTH) + P(THH) = 1/8 + 1/8 + 1/8 = 3/8 = .375
- p(3) = P(X = 3) = P(HHH) = 1/8 =.125
- In formula form:
- p(x) = (3Cx)(1/8), x=0,1,2,3.
- In tabular and graphical forms the probability function can be presented via:

Properties and Utilities of Probability Functions

- Note that the sum of the probabilities, which are all nonnegative, in a probability function equals 1. This is so because we take into account all the possible outcomes, that is, the sample space, and its probability is 1. That is, p(x) satisfies
- p(x) > 0 for all x, and all x {p(x)} = 1.
- The shape of the distribution could be obtained from the graph of the probability function.
- For example, the probability distribution for the variable X is symmetric with high points at the values of 1 and 2.

Utilities … continued

- The probability function of a random variable serves as a theoretical model of the population of values of the variable, with this population being the collection of the outcomes of the experiment when it is repeated many, many, many times.
- Numerical characteristics of the probability function, such as the mean, variance, and standard deviation, will be called parameters.
- The probability function can be used to compute the probabilities that the variable takes values in a certain set of interest.
- For instance, in the example, P(X < 2) = p(0) + p(1) + p(2) = 1/8 + 3/8 + 3/8 = 7/8.

Parameters of a Discrete Probability Distribution

- As mentioned earlier, a probability distribution serves as a theoretical model of a population.
- Characteristics of a probability distribution are therefore called parameters, and they are usually denoted by Greek letters.
- We now discuss four important parameters of discrete probability distributions. These are:
- Mean () and Median ()
- Variance (2) and Standard Deviation ().

Median of a Discrete Random Variable

- Given a discrete random variable X whose probability distribution function is p(x), its median, denoted by , is any value such that

A Simple Example

- For the variable X denoting the number of “H” that occur in the toss of three fair coins, we therefore have:
- Mean = = (0)(1/8)+(1)(3/8)+(2)(3/8)+(3)(1/8) = 12/8 = 1.5.
- For its median, we could take = 1.5, since notice that P(X < 1.5) = p(0) + p(1) = 1/8 + 3/8 = 0.5, and also, P(X > 1.5) = p(2) + p(3) = 1/8 + 3/8 = 0.5.
- However, note that the median is not unique since any value of between 1 and 2, exclusive, will satisfy the definition of being a median.

Interpretations

- As in the case of the sample statistics we have the following interpretations:
- The mean, , serves as the “center of gravity” or “balancing point” for the probability distribution; while
- The median is a value that divides the probability distribution into a 50:50 split.
- Other properties, like sensitivity of the mean to extreme values also holds for these parameters.

Variance of a Discrete Random Variable

- Given a discrete random variable X taking values {x1, x2, x3, …} whose probability distribution is p(x), its variance, denoted by 2, is given by:

Variance … continued

- The first formula is called the definitional formula, which indicates that the variance is the mean of the squared deviations from the mean ().
- The second formula is called the computational or the machine formula, and is easier to implement in practice.
- The variance is always nonnegative, and becomes zero if and only if the random variable takes only one value (we say in this case that it is a degenerate variable). The larger the value of the variance, the more variability in the distribution.
- The variance has squared units of measurements.

Standard Deviation of a Discrete Random Variable

- Since the variance has squared units of measurements, to obtain a measure of variation whose units are the same as the variable, we define the standard deviation () to be the positive square root of the variance.
- Formally, it is defined via:

Illustration of Computation of the Variance and Standard Deviation

- Going back to the variable X which counts the number of “H” in a toss of three fair coins, we have, by recalling that = 1.5 and using the definition, that:
- 2 = Var(X) = (0 - 1.5)2(1/8) + (1 - 1.5)2(3/8) + (2 - 1.5)2(3/8) + (3 - 1.5)2(1/8) = (2.25)(.125) + (.25)(.375) + (.25)(.375) + (2.25)(.125) = 0.75.
- We could also use the computational formula to get:
- 2 = Var(X) = [(0)2(.125) + (1)2(.375) + (2)2(.375) + (3)2(.125)] - (1.5)2 = [0 + .375 + 1.5 + 1.125] - 2.25 = 3 - 2.25 = 0.75.
- Therefore, = StdDev(X) = (.75)(1/2) = .866.

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