Homework 3

Question 1

Verify f is flow and compute its value

Answer 1

Show that f(e)<=w(e) (where f is the function given and w(e) is the given on the graph
Show non trivial notes at equal (just put f(SC) = all other f with same value=value

Flow = all values out of source/into sink

Question 2

If a game is equal to zero what does that mean

Answer 2

it is fair :)

Question 3

when is a game stable?

Answer 3

when the number of each best strategies of each player is equal

Question 4

identify a f augmented path and compute capacity

Answer 4

any path that is not equal to zero when w(e) - f(e) and capacity is the lowest number (number cannot be negative as there is no flow??)

Question 5

expected value of exponential distribution with parameter a

Answer 5

1/a

Question 6

variance of of exponential distribution with parameter a

Answer 6

1/(a^2)

Question 7

expected value of poisson distribution with parameter at

Answer 7

at

Question 8

variance of of poisson distribution with parameter a

Answer 8

at

Question 9

Let 𝑉 and 𝑊 be independent copies of 𝑇. Compute the quantities 𝑃[1 ≤𝑇 ≤ 3|𝑉 ≤ 3] and 𝑃[𝑇 + 𝑉 + 𝑊 ≤ 1]

Answer 9

Since 𝑇 and 𝑉 are independent, [+2]𝑃[1 ≤ 𝑇 ≤ 3|𝑉 ≤ 3] = 𝑃[1 ≤ 𝑇 ≤ 3] = 𝑃[𝑇 ≤ 3] − 𝑃[𝑇 ≤ 1]= (1 − exp(−1 ∗ 3)) − (1 − exp(−1 ∗ 1)) = exp(−1) − exp(−3) ≈ 0.32 The sum 𝑇 + 𝑉 + 𝑊 has Erlang distribution with parameters (3,1). Therefore, 𝑃[𝑇 + 𝑉 + 𝑊 ≤ 1] = 1 − ∑from k= 0 to 2 ( (1 ∗ 1)𝑘 exp(−1 ∗ 1))/𝑘! = 1 − 5/2 *exp(−1)

Question 10

P[T<=t] for exponential function

Answer 10

P[T<=t] = 1 - exp(-at)

Question 11

P[T>t] for exponential function

Answer 11

exp(-at)