Discrete probability distributions are used to determine the the probability of a specific event occurring. Meteorologists use discrete probability distributions to predict the weather, gamblers use them to predict the toss of the coin and financial analysts use them to calculate the probability of returns on their investments. The calculation of a discrete probability distribution requires that you construct a three-column table of events and probabilities, and then construct a discrete probability distribution plot from this table.
Make a probability distribution table for the weather. First assign all rainy days, the variable 1; all cloudy days, the variable 2; and all sunny days the variable 3. Now draw a table with three columns and three rows. Enter 1 in the first row in the first column, for rainy days; enter 2 in the second row of the first column for cloudy days; and enter 3 in the third row of the first column for sunny days.
Now pick a month with 31 days and find out how many rainy days, how many cloudy days and how many sunny days were in that month. If you don't have weather data, use 12 rainy days, 6 cloudy days and 13 sunny days. Note that 12 plus 6 plus 13 adds to 31, the number of days in the month.
Calculate the probability of each event. Divide the number of occurrences of a specific event by the total number of events. For this example, consider that 31 is the total number of events and the probability of a rainy day is calculated by dividing 12 by 31, to obtain 12/31. Similarly, the probability of a cloudy day is 6/31 and the probability of a sunny day is 13/31. Note that the sum of the probabilities equals 1, as it should. Convert these fractions to decimals. You should obtain 0.39, 0.19, and 0.42. In the third column of each row enter in these calculated probabilities in the same row as the associated events. 0.39 should be in the first row of the third column, 0.19 should be in the second row of the third column and 0.42 should be in the third row of the third column.
Now label the second column, x, and the third column, y.
Plot the discrete probability distribution. Make a coordinate x-y system on your graph paper. For this example, mark each grid mark on the graph paper on the x-axis using increments of 1, from 0 to 3. Make each grid mark on the y-axis using increments of 0.1, from 0 to 1.0. For each weather variable, that is 1, 2 and 3, in the x-column, and the corresponding probability calculated, in the y-column, plot the corresponding x, y coordinates. That is plot (1, 0.39), (2, 0.19) and (3, 0.42).
Now draw a vertical line from each of these points to the x-axis. This is your discrete probability distribution for the weather for the month.
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