Appendix D

Statistics

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Lab Contents

Probability - Introduction

An experiment or process is called random if on a given instance we cannot control or predict which one of all its possible outcomes will occur. Examples of random experiments are the rolling of dice, sex at conception (male or female), radioactive decay and measurement error due to human judgment. Any variable whose value depends upon the outcome of a random experiment is called a random variable.


Example: Dice

Suppose that two dice are rolled and we define the variable X to be the sum of spots on the two dice. X is a random variable since it depends on the outcome of a random experiment. Some possible outcomes are;

first die  1   3   2   5   4   6 
second die   1  2   6   5   3   6 

sum X  2   5   8   10  7   12
 Notice that X can have any integral value between 2 and 12 inclusive.

Example: Radioactive Decay

Radioactive decay is an example of a basic, real world statistical problem. For a radioactive substance we can define X as the number of particles that decay in a five second interval. X is a random variable since it depends on the outcome of a random process. Note that X will have integer values and is positive.

In this random process we have defined a random variable (X = counts in 5 sec.) but we do not know the probability distribution of the random variable. Thus we do not know the values that the random variable may have nor do we know any corresponding probabilities. In this situation, we analyze data from a large number of trials with the expectation that this will represent the values of the random variables, estimates of their probabilities and the shape of the probability distribution. The "law of large numbers" states that "as one increases the number of times an experiment is repeated then the theoretical probability of an event is approached by the ratio of occurrences of that event to the number of trials".


The two examples illustrate an important difference between them. In the dice example, all outcomes are known and their probabilities can be calculated. In the radioactivity example, the outcomes or their probabilities are not known.