As the name suggests, the independent variable (or random variable) is the physical event from which we derive our predictions, and the dependent variable (or random number) is the measurement taken from this event. For example, if we wish to determine the distribution of the random numbers drawn at random at the beginning of each game played at a regular interval between players, we would first have to determine the start time of the game and the length of the interval. Then, taking the value of the rate of change of temperature of water in the bucket measured at the two previous times we measure the temperature, we can calculate the probability that at the end of each period (i.e., at the end of the game) the temperature of the water in the bucket will be measured at a certain value. Using this idea, we would then estimate the value of the independent variable for the time interval from the time we measure the temperature to the end of the game and multiply this estimate by the value of the dependent variable. Now we can determine the probability that at the end of the game, the temperature of the water in the bucket will be measured at a certain value. This procedure, however, is not always applicable to a game that begins after a delay of time: there could be situations where the end of the game is earlier than the measurement of the temperature.
It is possible, however, to combine the idea of the random number with the idea of the independent variable in such a way as to make the probability of a measurement at some particular time higher than the probability of the measurement at a later time. We could, for example, take the temperature of the water in the bucket at one of the last measurements and determine a series of random numbers drawn at different times of the day, using a stochastic random number generator and then calculate the probability that the temperature of the water in the bucket will be measured at the same value at some later time. There would, of course, be several ways in which this could happen. First, the temperature might go up during the time of the measurement, and the temperature of the bucket might go down during the same time. Second, the temperature may remain constant throughout the time of the measurement, but go down when the measurement is taken next. Third, there may be fluctuations in the temperature of the bucket itself; for example, it may rise slightly during the measurement, but fall a little in the next measurement.
The frequency at which these fluctuations take place will affect the random number generator, and the frequency at which the generator has to stop working in order to maintain the same value of the independent variable. These fluctuations, due to the Law of Independence, will depend only on the initial value of the independent variable and the condition of the random number generator, and therefore the frequency at which the temperature of the water in the bucket is measured will depend on the initial value and on the frequency at which the random number generator has to stop working.
Here is another simple example: Suppose that we have two players playing in a football game, one of whom scores four goals in each game. At the end of the game, the team which scores more goals wins the game. The second player, with the fewest goals, scores the winning goal, but he still needs to play another game to score his four goals in the other game.
If the two teams were equally matched in strength, it would be impossible to tell which team had been more likely to win simply by comparing the statistics of the two players with the statistics of the other players, since there would be fluctuations between the players that will affect the statistics. according to the way in which they have played against each other.
Therefore, we need to know whether it was the team that won, or the player who scored a goal, or the other team that scored the most goals, that was more likely to win, by looking at the statistics of the team with which each player had the best statistical record. This will depend on whether the player was playing for the same team at the end of the game. If the player is playing for a team where the goal difference is small, his statistics will also differ according to the results of the previous game, while if he plays for a team where the goal difference is large, his statistics will differ based on whether he is playing for the losing team or the winning team.
This situation is similar to an example of the Law of Independence: we can compare the player’s performance against another player who plays for a losing team with another player’s performance against a team in which the goal difference is large. We can compare the statistics for each player on the same team with a statistic for a player who plays on a team in which the goal difference is small. If the player whose statistics are different is playing for the losing team is a good player, but the player whose statistics are the same is not, we will say that he is the bad player, because his statistics will differ between the two games, and his performance will be affected by the results of the previous game. Thus, the Law of Independence is also applicable to random variables.