Combining and Subtracting Distributions

I. Combining Distributions

Let's say that Theresa's time for running a mile is normally distributed with a mean of 7 minutes and standard deviation of 0.2 minutes, and Shannon's time for running a mile is normally distributed with a mean of 6 minutes and standard deviation of 0.4 minutes.

T~N(7, 0.2)

S~N(6, 0.4)

They are running a 2 mile relay race, so you want to know what their combined distribution looks like (we'll call this distribution "R" for relay race). In other words, you want to fill in the following blanks: R~N(___ , ___)

Here's a few rules to help:

COMBINING/ADDING DISTRIBUTIONS:  (1) Mean: $$\mu_A = \mu_X + \mu_Y$$     (2) Variance:  $$\sigma^2_A = \sigma^2_X + \sigma^2_Y$$      (3) Standard Deviation: $$\sigma_A = \sqrt{\sigma^2_A} = \sqrt{\sigma^2_X + \sigma^2_Y}$$

In other words, the mean of the combined distribution is found by ADDING the two individual means together. The variance of the combined distribution is found by ADDING the two individual variances together. The standard deviation is the square root of the variance. Note that the standard deviation is ONLY found by taking the square root of the combined variances; it is NOT found by combining the two individual standard deviations.

So, returning to the original example, let's find the mean and standard deviation of our combined distribution R.

Mean:  $$\mu_R = \mu_T + \mu_S$$  →   $$\mu_R = 7 + 6$$  →  $$\mu_R = 13$$

Standard Deviation:  $$\sigma_R = \sqrt{\sigma^2_T + \sigma^2_S}$$   →  $$\sigma_R = \sqrt{(0.2)^2 + (0.4)^2}$$   →  $$\sigma_R = \sqrt{0.2}$$  →  0.447

II. Subtracting Distributions

Let's say that you are running a business. You know that your income is normally distributed with a mean of $100 a day and a standard deviation of$10, and your expenses are normally distributed with a mean of $40 a day and a standard deviation of$5.

I~N(100, 10)

E~N(40, 5)

You want to find the daily net profit of running your business. Profits are calculated by subtracting expenses from income (i.e. profit = income - expenses). Thus, you need to find out what the subtracted distribution looks like (we'll call this distribution "P" for profit). In other words, you want to fill in the following blanks: P~N(___ , ___)

Here's a few rules to help:

SUBTRACTING DISTRIBUTIONS:  (1) Mean: $$\mu_S = \mu_X - \mu_Y$$     (2) Variance:  $$\sigma^2_S = \sigma^2_X + \sigma^2_Y$$      (3) Standard Deviation:  $$\sigma_S = \sqrt{\sigma^2_S} = \sqrt{\sigma^2_X + \sigma^2_Y}$$

In other words, the mean of the combined distribution is found by SUBTRACTING the two individual means from each other. Now, note how the variance of the combined distribution is found by ADDING the two individual variances together. Even though we are subtracting the distributions, variability is still increasing, so we still add the variances together.

So, returning to the original example, let's find the mean and standard deviation of our subtracted distribution P.

Mean:  $$\mu_P = \mu_I - \mu_E$$   →  $$\mu_P = 100 - 40$$  →  $$\mu_P = 60$$

Standard Deviation: $$\sigma_P = \sqrt{\sigma^2_I + \sigma^2_E}$$  →  $$\sigma_P = \sqrt{(10)^2 + (5)^2}$$  → $$\sigma_P = \sqrt{125}$$   →  $$\sigma_P = 11.18$$

III. Summary

Mean Variance

$$\mu_A = \mu_X + \mu_Y$$ $$\sigma^2_A = \sigma^2_X + \sigma^2_Y$$
$$\mu_S = \mu_X - \mu_Y$$ $$\sigma^2_S = \sigma^2_X + \sigma^2_Y$$