WARM – UP
Refer to the square diagram below and let X be the xcoordinate and Y be the y-coordinate of any randomly chosen
point. Find the Conditional Probability P(Y < ½ | Y > X) = ?
Y
1
P(Y < ½ | Y > X) =
0.5
P(Y < ½ | Y > X) =
0
P(Y < ½ ∩ Y > X)
P(Y > X)
X
0
0.5
1
1/8
1/2
1/4
VALID WAYS OF PROVING INDEPENDENCE
P(A∩B) = P(A)∙P(B)
P(A) = P(A|B) = P(A|C) = P(A|D)
X2 Test of Independence
INVALID WAYS OF PROVING INDEPENDENCE
P(A∩B) = P(A)
P(A|B) = P(B|A)
P(A|C) = P(B|C)
P(A) = P(B)
P(A∩B) = P(AUB)
P(A∩B) = P(AUC)
CHAPTER 16 - MEANS OF RANDOM VARIABLES
The Mean of a random variable is a weighted average of the
possible values of X. The Mean is also called the Expected
Value and is noted by the symbol, μx or E(X)
Values of X
x1
x2
x3
…
xk
Probability
p1
p2
p3
…
pk
The MEAN of a DISCRETE VARIABLE (Weighted Average)
Let X = the random variable whose distribution follows:
μx = x1p1 + x2p2 + x3p3 + … + xkpk
E(X) = μx = Σxi pi
EXAMPLE #1:
An insurance company acknowledges that its
payout probability for claims is as follows:
1. What can the company
expect to payout per
policyholder?
2. If they insure 200 clients
and each client pays $100
for the policy, how much
profit should they expect?
Policyholder
Outcome
Policy
x
Probability
P(X = x)
Death
$10000
1 / 1000
Disability
$5000
2 / 1000
$0
997 / 1000
Neither
1. E(X) = μx = x1p1 + x2p2 + x3p3
=
Σxi·pi
E(X) = μx = 10000(1/1000) + 5000(2/1000) + 0(997/1000) = $20
2. Profit = Revenue – Expenses
= 200($100) – 200($20) = $16000
The following probability distribution represent
the AP Statistics scores from previous years.
AP Score
1
2
3
4
5
Probability 0.14 0.24 0.34 0.21 0.07
EXAMPLE #2:
What can the average student expect to make on the AP Exam?
μx = E(X) = 1(.14) + 2(.24) + 3(.34) + 4(.21) + 5(.07) =
μx = E(X) = 2.83
The VARIANCE OF A RANDOM VARIABLE
The Variance and the Standard Deviation are the measures of
the spread of a distribution. The variance is the average of the
square deviations of the variable X from its mean (X – μx)2 .
The variance is denoted by: σX2. The Standard Deviation is the
square root of the Variance and is denoted by: σx.
Let X = the random variable whose distribution follows and has
mean :E(X) = μx = Σxi·pi
Values of X
x1
x2
x3
…
xk
Probability
p1
p2
p3
…
pk
The Variance of a discrete random variable is:
σx2 = (x1 – μx)2 p1 + (x2 – μx)2 p2 + … + (xk – μx)2 pk
Var(X) = σx2 = Σ (xi – μx)2 pi
The following probability distribution represent s
the AP Statistics scores from previous years.
AP Score
1
2
3
4
5
Probability 0.14 0.24 0.34 0.21 0.07
EXAMPLE #2:
What is the Standard Deviation of the AP Exam results?
μx = E(X) = 1(.14) + 2(.24) + 3(.34) + 4(.21) + 5(.07) =
μx = E(X) = 2.83
σx2 = (x1 – μx)2 p1 + (x2 – μx)2 p2 + … + (xk – μx)2 pk
= (1 – 2.83)2·(.14) + (2 – 2.83)2·(.24) + (3 – 2.83)2· (.34) + (4 –
2.83)2·(.21) + (5 – 2.83)2· (.07)
= Σ (xi – μx)2 pi
σx2 = 1.2611
σx = 1.123
EXAMPLE #1:
An insurance company acknowledges that its
payout probability for claims is as follows:
1. What can the company
expect to payout per
policyholder?
Policyholder
Outcome
2. What is the Standard
Deviation of payouts for
the distribution of
Policyholders?
Policy
x
Probability
P(X = x)
Death
$10000
1 / 1000
Disability
$5000
2 / 1000
$0
997 / 1000
Neither
1. E(X) = μx = 10000(1/1000) + 5000(2/1000) + 0(997/1000) = $20
2. σx2 = (x1 – μx)2 p1 + (x2 – μx)2 p2 + … + (xk – μx)2 pk
= (10000 – 20)2·(1/1000) + (5000 – 20)2·(2/1000) + (0 – 20)2· (997/1000)
σx2 = 149600
σx = √149600 = $386.78
The reason why you SHOULD NOT gamble!
Expected Payouts
on a $1.00 Bet
PAYOUTS
Color
1 to 1
Single #
35 to 1
E(X) = μx = xWINpWIN + xLOSEpLOSE
E(RED) = μ = $1(18/38) + -$1(20/38)
μ = $–0.05
E(25) = μ = $35(1/38) + -$1(37/38)
μ = $–0.05
The reason why you SHOULD NOT gamble!
Expected Payouts
on a $1.00 Bet
PAYOUTS
Corner
8 to 1
Split
17 to 1
E(X) = μx = xWINpWIN + xLOSEpLOSE
E(Corner) = μ = $8(4/38) + -$1(34/38)
μ = $–0.05
E(Split) = μ = $17(2/38) + -$1(36/38)
μ = $–0.05
The reason why you SHOULD NOT gamble!
Expected Payouts
on a $1.00 Bet
PAYOUTS
Street
11 to 1
Dozen
2 to 1
E(X) = μx = xWINpWIN + xLOSEpLOSE
E(Street) = μ = $11(3/38) + -$1(35/38)
μ = $–0.05
E(Dozen) = μ = $2(12/38) + -$1(26/38)
μ = $–0.05