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A Random Variable over a sample space is a function that maps every sample point or outcome to a real number. Numerical values are assigned to outcomes in many experiments. A random variable is a function that assigns a value, essentially a numeric value to all possible outcomes of a random experiment.
These numbers are called values of the random variable. These are often denoted by using X or Y. A random variable is not a variable but rather a function that maps events or outcomes to numbers.
Suppose, if a coin is tossed three times in a row and the possible outcomes are noted, then the possible outcomes for the experiment will be S = {HHH, HHT, HTH, HTT, THH, TTT}. If X is taken as the random variable to the number of heads, then SX = {0, 1, 2, 3}.
Discrete Random Variables
A discrete random variable can take only finite, specific, isolated numeric values, like the outcome of a roll of dice, or the number of dollars in a randomly chosen bank account.
Continuous Random Variables
A continuous random variable on the other hand can take infinite values, like the temperature of a room for one continuous day or the height of a man in centimeters.
Sometimes, it is necessary to add or subtract random variables. When this occurs, it is useful to know the mean and variance of the result.
Sums and Differences of Random Variables: Effect on the Mean
Suppose you have two variables: X with a mean of μx and Y with a mean of μy. Then, the mean of the sum of these variables μx+y and the mean of the difference between these variables μx-y are given by the following equations.
μ_{x+y} = μ_{x}_{ }+ μ_{y } and μ_{x-y} = μ_{x} - μ_{y}
The above equations for general variables also apply to random variables. If X and Y are random variables, then
E(X + Y) = E(X) + E(Y) and E(X - Y) = E(X) - E(Y)
where E(X) is the expected value (mean) of X, E(Y) is the expected value of Y, E(X + Y) is the expected value of X plus Y, and E(X - Y) is the expected value of X minus Y.
Independence of Random Variables
If two random variables, X and Y, are independent, they satisfy the following conditions.
ï‚§ P(x|y) = P(x), for all values of X and Y.
ï‚§ P(x ∩ y) = P(x) * P(y), for all values of X and Y.
The above conditions are equivalent. If either one is met, the other condition also met; and X and Y are independent. If either condition is not met, X and Y are dependent.
Note: If X and Y are independent, then the correlation between X and Y is equal to zero.
Sums and Differences of Random Variables: Effect on Variance
Suppose X and Y are independent random variables. Then, the variance of (X + Y) and the variance of (X - Y) are described by the following equations
Var(X + Y) = Var(X - Y) = Var(X) + Var(Y)
where Var(X + Y) is the variance of the sum of X and Y, Var(X - Y) is the variance of the difference between X and Y,Var(X) is the variance of X, and Var(Y) is the variance of Y.
Note: The standard deviation (SD) is always equal to the square root of the variance (Var). Thus,
SD(X + Y) = sqrt[ Var(X + Y) ] and SD(X - Y) = sqrt[ Var(X - Y) ]
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