RVCND - Combining Discrete Random Variables Lesson
Combining Discrete Random Variables Lesson
Independence of two events (variables) continues to be important. An example of two independent events is rolling a die and tossing a coin. The outcome of one event does not affect the outcome of the other. Summary statistics including mean and standard deviation of independent random variables may need to be combined. We learned earlier that adding or subtracting a constant value from each data point simply "shifts" the mean along the horizontal axis but does not change the variance or standard deviation (the spread). The same is true for random variables and yields these formulas:
E(X + c) where c is some constant is defined as E(X) + c.
Var (X + c) is defined as Var (X) + c.
Remember we use the term "expected value" to suggest the mean.
EXAMPLE: If everyone in a math class had 5 points added to a particular test grade, the NEW class average on that test would be 5 points higher than the original mean. If the standard deviation were 3 points on the original test, it would remain 3 points on the NEW test since spread is not affected by addition.
Multiplication rules are a little different. In general, multiplying each value of a random variable by a constant multiplies the mean by that constant and the variance by the square of the constant.
E(aX) = (a)E(X)
Var(a(X)) = a2 Var(X)
Considering two independent random variables, the expected value of the sum or difference is the sum or difference of the expected values.
E(X + Y) = E(X) + E(Y)
Combining standard deviations is done through working with the variance values which add, or subtract, mathematically. Since standard deviations are values obtained from square roots they only combine under special circumstances making it is NECESSARY to work with the variances.
Var (X + Y) = Var (X) + Var (Y)
NOTE: BE AWARE THAT COMBINED RANDOM VARIABLES FORMED BY EITHER ADDING OR SUBTRACTING BOTH RESULT IN ADDITION OF THE VARIANCES.
Then square root to determine the standard deviation value.
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