Explain the effect of adding a constant to a random variable.
Shifts the mean by the constant; standard deviation remains unchanged.
Explain the effect of multiplying a random variable by a constant.
Multiplies both the mean and the standard deviation by the constant (absolute value for SD).
Explain how to find the mean of the sum of two random variables.
The mean of the sum is the sum of the means: E(X + Y) = E(X) + E(Y).
Explain how to find the mean of the difference of two random variables.
The mean of the difference is the difference of the means: E(X - Y) = E(X) - E(Y).
Explain how to find the standard deviation of the sum or difference of two *independent* random variables.
First, find the variance of the sum/difference by adding the variances. Then, take the square root to find the standard deviation.
Why do we use variances when combining random variables?
Variances represent the 'energy' of the random variables, which add up when combining them. Standard deviations don't add directly.
Define 'random variable'.
A variable whose value is a numerical outcome of a random phenomenon.
What is 'expected value'?
The mean of a random variable; the long-run average outcome.
Define 'variance'.
A measure of the spread of a random variable's distribution; the average squared deviation from the mean.
Define 'standard deviation'.
The square root of the variance; a measure of the typical deviation of a random variable from its mean.
What is a 'linear transformation'?
Transforming a random variable by multiplying it by a constant and/or adding a constant.
If Y = X + c, what is E(Y)?
E(Y) = E(X) + c
If Y = X + c, what is SD(Y)?
SD(Y) = SD(X)
If Y = c * X, what is E(Y)?
E(Y) = c * E(X)
If Y = c * X, what is SD(Y)?
SD(Y) = |c| * SD(X)
If S = X + Y, what is E(S)?
E(S) = E(X) + E(Y)
If D = X - Y, what is E(D)?
E(D) = E(X) - E(Y)
If S = X + Y, what is Var(S)?
Var(S) = Var(X) + Var(Y)
If D = X - Y, what is Var(D)?
Var(D) = Var(X) + Var(Y)
If S = X + Y, what is SD(S)?
SD(S) = \(\sqrt{Var(X) + Var(Y)}\)
If D = X - Y, what is SD(D)?
SD(D) = \(\sqrt{Var(X) + Var(Y)}\)
