Sanderson M. Smith

(BUT STANDARD DEVIATIONS CAN'T)

Here is an illustration of these important statistical formulas:

If X and Y are any two random variables, then

mX+Y = mX + mY

If X and Y are independent random variables, then

s2X+Y = s2X + s2Y

Note: These "nice" rules do not hold for standard deviations.

Here's a story about my friend, Herkimer, to illustrate what is stated above:

Herkimer works only on weekends. On Saturday there are three possible financial outcomes for him, each with probability = 1/3 = 33 1/3% He can end up the day \$100 in debt, he can end the day up with a gain of \$50, or he can end up the day with a gain of \$200. On Sunday, he has a similar situation, although on that day there are five possible financial outomes, each with probability = 1/5 = 20%. The following table shows Herkimer's possible gains and losses for Saturday and Sunday, along with the set (Sa + Su), containing the 5x3 = 15 gains and losses he can have at the end of a weekend.

 Sa (Saturday) Su (Sunday) (Sa + Su) (Saturday+Sunday) -\$100 -\$80 -\$180 \$50 -\$40 -\$140 \$200 \$25 -\$75 \$90 -\$10 \$160 \$60 ------------ ------------ -\$30 Means \$50.00 \$31.00 \$10 Standard Deviations \$122.47 \$86.63 \$75 Variances (sq. dollars) \$15,000.00 \$7,504.00 \$140 \$210 \$120 \$160 \$225 \$290 \$360 ------------ Mean(Sa) + Mean (Su) \$81.00 <------MEANS ADD--------> \$81.00 <--Mean(Sa+Su) St.Dev.(Sa)+ St.Dev.(Su) \$197.47 <---STANDARD DEVIATIONS DO NOT ADD--> \$150.01 <--St. Dev. (Sa+Su) Var(Sa) + Var(Su) \$22,504.00 <---VARIANCES ADD---> \$22,504.00 <--Variance(Sa+Su)