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Mean and Standard Deviation of Random Variables
This note page expands on the concepts introduced in Introduction to Random Variables and Probability Distributions, focusing specifically on quantifying the center and spread of a random variable’s distribution.
Expected Value (Mean) of a Discrete Random Variable
The expected value or mean of a discrete random variable $ X $ , denoted as $ E(X) $ or $ \mu_X $ , represents the long-run average value of the variable over many repetitions of the random process. It is a measure of the center of the probability distribution.
To calculate the expected value of a discrete random variable, you multiply each possible value of the variable by its corresponding probability and then sum these products.
$$ \mu_X = E(X) = \sum x_i p_i
$$
Where:
- $ x_i $ are the individual values the random variable $ X $ can take.
- $ p_i $ are the probabilities associated with each $ x_i $ .
Example: Expected Value
Consider a game where you roll a fair six-sided die. If you roll a 6, you win \ $ 5. If you roll a 1 or 2, you lose \ $ 2. Otherwise, you neither win nor lose. Let $ X $ be the amount of money you win/lose.
| $ x_i $ (Outcome) | $ p_i $ (Probability) |
|---|---|
| \ $ 5 | $ 1/6 $ |
| -\ $ 2 | $ 2/6 $ |
| \ $ 0 | $ 3/6 $ |
The expected value is: $$ E(X) = (5 \times \frac{1}{6}) + (-2 \times \frac{2}{6}) + (0 \times \frac{3}{6}) \ E(X) = \frac{5}{6} - \frac{4}{6} + 0 = \frac{1}{6} \approx \ $ 0.17 $$ On average, you can expect to win about \ $ 0.17 per game in the long run.
Standard Deviation of a Discrete Random Variable
The standard deviation of a random variable, denoted as $ \sigma_X $ , measures the typical distance of the values of the variable from the mean. It quantifies the spread or variability of the distribution. A larger standard deviation indicates greater spread.
Before calculating the standard deviation, we first find the variance, denoted as $ Var(X) $ or $ \sigma_X^2 $ . The variance is the expected value of the squared deviations from the mean.
$$ \sigma_X^2 = Var(X) = \sum (x_i - \mu_X)^2 p_i
$$
The standard deviation is simply the square root of the variance:
$$ \sigma_X = \sqrt{\sum (x_i - \mu_X)^2 p_i} $$
Example: Standard Deviation
Using the previous die roll example, where $ \mu_X = 1/6 $ :
| $ x_i $ | $ p_i $ | $ (x_i - \mu_X) $ | $ (x_i - \mu_X)^2 $ | $ (x_i - \mu_X)^2 p_i $ |
|---|---|---|---|---|
| \ $ 5 | $ 1/6 $ | $ 5 - 1/6 = 29/6 $ | $ (29/6)^2 = 841/36 $ | $ (841/36) \times 1/6 = 841/216 $ |
| -\ $ 2 | $ 2/6 $ | $ -2 - 1/6 = -13/6 $ | $ (-13/6)^2 = 169/36 $ | $ (169/36) \times 2/6 = 338/216 $ |
| \ $ 0 | $ 3/6 $ | $ 0 - 1/6 = -1/6 $ | $ (-1/6)^2 = 1/36 $ | $ (1/36) \times 3/6 = 3/216 $ |
Summing the last column gives the variance: $$ Var(X) = \frac{841}{216} + \frac{338}{216} + \frac{3}{216} = \frac{1182}{216} \approx 5.472 $$ The standard deviation is: $$ \sigma_X = \sqrt{\frac{1182}{216}} \approx \sqrt{5.472} \approx \ $ 2.34 $$ This means that, on average, the amount won or lost per game typically varies by about \ $ 2.34 from the expected win of \ $ 0.17.
Combining Random Variables
When dealing with multiple random variables, we often need to understand the mean and standard deviation of their sum or difference. This topic, known as Combining Random Variables, covers rules for linear transformations of random variables ( $ Y = a + bX $ ) and for combining independent random variables (e.g., $ W = X + Y $ or $ D = X - Y $ ). Key principles include:
- Mean: Means are additive for both sums and differences ( $ E(X \pm Y) = E(X) \pm E(Y) $ ).
- Variance: Variances are additive for the sum or difference of independent random variables ( $ Var(X \pm Y) = Var(X) + Var(Y) $ ). Note that standard deviations are not directly additive.
- Independence: The rule for adding variances only applies if the random variables are independent. If they are not independent, additional considerations related to covariance are needed.