19. 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?
21. What is the probability that
max(X1, X2 , X3 , X4 , X5 , X6 , X7) ≤ $100 ?
min(X1, X2 , X3 , X4 , X5 , X6 , X7) ≤ $ -100 ?
22. Recall from the univariate case, we have
two methods of calculating probabilities of
transformed variables
Distribution Change of
function variable
technique technique
24. Suppose the Xi are iid. Is this a reasonable
assumption?
Then, we can calculate Fv(a) by
P(V ≤ a) = P(min(Xi) ≤ a)
25. 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)
26. 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)
28. = 1 – [P(X1 > a, X2 > a, … X7 > a)]
= 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
29. = 1 – [P(X1 > a, X2 > a, … X7 > a)]
= 1 – [P(X1 > a) P(X2 > a) … P(X7 > a) ]
(Because the Xi are independent)
30. = 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) ]
31. = 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)
32. = 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 ]
33. = 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 ]
34. = 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 ]
35. So P(V ≤ -100) = Fv(-100) = 1 – [ (1 – Fx(-100) ) 7 ]
We can find the density of V by differentiating:
fv(a) = Fv(a)
36. 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 ]}
37. 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))
38. 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)