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b: Distinguish between and give examples of discrete and continous random variables.
A discrete random variableis one where we can list all the possible outcomes and for each possible outcome there is a measurable, positive probability. An example of a discrete random variable is the number of days it rains in a given month. A continuous distribution would define the probabilities for the actual amount of rainfall. A continuous random variableis one where we cannot list the possible outcomes because we can always list a third number between any two numbers on the list. The number of outcomes is essentially infinite even if lower and upper bounds exist. The number of points between the lower and upper bounds are essentially infinite.
c: Describe the range of possible outcomes of a specified random variable.
In the discrete case, such as the number of days of rain in a month, there are a finite number of outcomes as defined by the number of days in the month. In the continuous case, such as the amount of rainfall, the outcome can be recorded out to many decimal places. We say the probability of two inches of rainfall is essentially zero because it is a single point. We must talk about the probability of the amount of rain being between two and three inches. In other words:
For a discrete distribution p(x) = 0 when "x" cannot occur, or p(x) > 0 if it can.
For a continuous distribution p(x) = 0 even though "x" can occur, but we can only consider p(x1≤ X ≤ x2), where x1and x2are actual numbers.
