Transcript X - Mags Maths

Expected value
Question 1
• You pay your sales personnel a
commission of 75% of the amount they
sell over $2000. X = Sales has mean
$5000 and standard deviation $1000.
What are mean and standard deviation
of pay?
Question 1
• (X - 2000) represents the basis for the
commission
• and "Pay" is 75% of that
• Pay = (0.75)(X - 2000) = 0.75 X - 1500
• E[Pay] = E[ 0.75 X - 1500 ]
•
= 0.75 E[X] - 1500
•
= 0.75(5000) - 1500 = $2250
Question 1
• (X - 2000) represents the basis for the
commission
• and "Pay" is 75% of that
• Pay = (0.75)(X - 2000) = 0.75 X - 1500
• [Pay] =  [ 0.75 X - 1500 ]
•
= 0.75  [X]
•
= 0.75(1000)
•
= $750
• Note: We are not combining so we don’t need
to use variance.
Question 2 The Portfolio Affect
• You are considering purchase of stock in two
different companies, X and Y.
• Return after one year for stock X is a random
variable with X = $112, X = 10.
Return for stock Y (a different company) has
the same  and .
• Assuming that X and Y are independent,
which portfolio has less variability, 2 shares
of X or one each of X and Y?
Question 2 The Portfolio Affect
• The returns from 2 shares of X will be
exactly twice the returns from one
share, or 2X. The returns from one each
of X and Y is the sum of the two
returns, X+Y.
Question 2 The Portfolio Affect
E[ aX + b ]
= a E[X] + b
E[2X + 0] =
2(112) = 224
E[ aX+bY ] =
aE[X] + bE[Y]
E[ X+Y ]
= 112 + 112 = 224
Question 2 The Portfolio Affect
 aX b  a X
 2X  0

= 2(10) = 20

 aX bY
 a 2 2X + b2 Y2
 X Y  102 + 102
=14.14
Conclusion:
• X+Y has smaller standard deviation
than 2X.
Insight:
• Why does X+Y have a narrower probability
distribution than 2X?
• Since X and Y vary independently, losses in
one are sometimes offset by gains in the
other. With 2 shares of stock of the same
company, losses and gains are just doubled.
This is one version of the old saying, "Don’t
put all of your eggs into one basket!"
Question 3
• In what interval will the return of a
portfolio consisting of 2 units of stock X
and 3 units of stock Y occur 2/3 of the
time, according to the empirical rule?
(Use X and Y from question 2 i.e.
X = $112, X = 10. .)
Question 3
• First, use (3a) to get the mean:
EaX  bY

  

  
 a EX   b EY
 2 EX   3 EY



 $560

• Next, use "special case" formula (3b) to
get the standard deviation:
aX bY
 a2X2 + b2 Y2
 22102 + 32102


 $36.06
• The empirical rule states that
"approximately 2/3 of the time, a
random variable will be within 1 of its
mean".
• Here this interval is $560  36.06.
Question 4
• The selling price of a product is $30, but it
costs the seller $20. The forecast of the
number of units that will be sold in the
upcoming month is 5000, with standard
deviation 100. The seller has a fixed cost of
$8,000 per month. In what interval will net
profits lie for the upcoming month, with 95%
probability, according to the empirical rule?
The empirical rule states that "approximately
95% of the time, a random variable will be
within 2 of its mean".
Question 4
•
•
•
•
•
•
•
Let X = number of units sold next month.
Profits = (30 - 20)X - 8000 = 10X - 8000.
Expected profits = E[10X - 8000]
= 10(5000) - 8000 = $42,000.
Standard deviation of profits =  [10X - 8000]
= 10X = 10(100) = $1000.
The empirical rule states that "approximately
95% of the time, a random variable will be
within 2 of its mean" so the 95% range for
returns is $42,000  2(1,000).