9.
Your Turn
Suppose you own a portfolio of stocks. Let X1 be
the amount of money your portfolio earns
today, X2 be the amount of money it earns
tomorrow, and so on…
How would you calculate U and V, where U is
the amount of money you’ll make on your
best day during the next week, and V is the
amount you’ll make on your worst day?
11.
What is the probability that
max(X1, X2 , X3 , X4 , X5 , X6 , X7) ≤ $100 ?
min(X1, X2 , X3 , X4 , X5 , X6 , X7) ≤ $ -100 ?
12.
Recall from the univariate case, we have
two methods of calculating probabilities of
transformed variables
Distribution Change of
function variable
technique technique
14.
Suppose the Xi are iid. Is this a reasonable
assumption?
Then, we can calculate Fv(a) by
P(V ≤ a) = P(min(Xi) ≤ a)
15.
Suppose the Xi are iid. Is this a reasonable
assumption?
Then, we can calculate Fv(a) by
P(V ≤ a) = P(min(Xi) ≤ a)
= 1 – P(min(Xi) > a)
16.
Suppose the Xi are iid. Is this a reasonable
assumption?
Then, we can calculate Fv(a) by
P(V ≤ a) = P(min(Xi) ≤ a)
= 1 – P(min(Xi) > a)
= 1 – P(all Xi > a)
18.
= 1 – [P(X1 > a, X2 > a, … X7 > a)]
= 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
19.
= 1 – [P(X1 > a, X2 > a, … X7 > a)]
= 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
(Because the Xi are independent)
20.
= 1 – [P(X1 > a, X2 > a, … X7 > a)]
= 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
(Because the Xi are independent)
= 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ]
21.
= 1 – [P(X1 > a, X2 > a, … X7 > a)]
= 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
(Because the Xi are independent)
= 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ]
(because the Xi are identically distributed)
22.
= 1 – [P(X1 > a, X2 > a, … X7 > a)]
= 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
(Because the Xi are independent)
= 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ]
(because the Xi are identically distributed)
= 1 – [P(X1 > a) 7 ]
23.
= 1 – [P(X1 > a, X2 > a, … X7 > a)]
= 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
(Because the Xi are independent)
= 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ]
(because the Xi are identically distributed)
= 1 – [P(X1 > a) 7 ]
= 1 – [ (1 – P(X1 ≤ a) )7 ]
24.
= 1 – [P(X1 > a, X2 > a, … X7 > a)]
= 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
(Because the Xi are independent)
= 1 – [P(X1 > a) P(X1 > a) … P(X1 > a) ]
(because the Xi are identically distributed)
= 1 – [P(X1 > a) 7 ]
= 1 – [ (1 – P(X1 ≤ a) )7 ]
= 1 – [ (1 – Fx(a) ) 7 ]
25.
So P(V ≤ -100) = Fv(-100) = 1 – [ (1 – Fx(-100) ) 7 ]
We can find the density of V by differentiating:
fv(a) = Fv(a)
26.
So P(V ≤ -100) = Fv(-100) = 1 – [ (1 – Fx(-100) ) 7 ]
We can find the density of V by differentiating:
fv(a) = Fv(a)
= {1 – [ (1 – Fx(a) ) 7 ]}
27.
So P(V ≤ -100) = Fv(-100) = 1 – [ (1 – Fx(-100) ) 7 ]
We can find the density of V by differentiating:
fv(a) = Fv(a)
= {1 – [ (1 – Fx(a) ) 7 ]}
= -7(1 – Fx(a) ) 6 (1 - Fx(a))
28.
So P(V ≤ -100) = Fv(-100) = 1 – [ (1 – Fx(-100) ) 7 ]
We can find the density of V by differentiating:
fv(a) = Fv(a)
= {1 – [ (1 – Fx(a) ) 7 ]}
= -7(1 – Fx(a) ) 6 (1 - Fx(a))
= 7(1 – Fx(a) ) 6 fx(a)
29.
Your Turn
Work through the handout to find FU(a) and
fU(a).
30.
What if we wish to find the joint distribution
FU,V(a,b)?
U = max(X, Y)
V = min(X, Y)
P(U < 2, V < 5) = ?
31.
Probability as volume under a surface
f(x,y)
P(Set A)
X
Set A
Y
32.
P(U < 2, V < 5) = P( max(X, Y) < 5 min(X, Y) > 2)
f(x,y)
P(Set A)
X
Set A
Y
5 5
P(U < 2, V < 5) = ∫ ∫ fx,y (x,y) dx dy
2 2
33.
But…
•Computing double integrals can
be hard
•Finding correct bounds can be
hard
r = √((x - 25)2 + y2 )
Ө = tan-1 (y/(x – 25))
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