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What are the differences between adding a constant and multiplying by a constant in transforming a random variable?
Adding: Shifts the center, no change to spread. | Multiplying: Changes both the center and the spread.
What are the differences between the effects on variance and standard deviation when combining random variables?
Variance: Variances are added together. | Standard Deviation: Standard Deviation is the square root of the added variances.
What are the differences between calculating the mean of a sum versus the standard deviation of a sum of random variables?
Mean of Sum: Add the means directly. | Standard Deviation of Sum: Add the *variances*, then take the square root.
What are the differences between calculating the mean of a difference versus the standard deviation of a difference of random variables?
Mean of Difference: Subtract the means directly. | Standard Deviation of Difference: Add the *variances*, then take the square root.
What are the differences between standard deviation and variance?
Standard deviation: Measure of dispersion expressed in the same units as the data. | Variance: Measure of dispersion expressed in squared units.
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) = \(\sqrt{Var(X) + Var(Y)}\)
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.