Probability

Posted On February 23, 2018 at 2:29 am by / Comments Off on Read e-book online Advanced Level Mathematics: Statistics 2 PDF

By Steve Dobbs, Jane Miller, Julian Gilbey

ISBN-10: 0521530148

ISBN-13: 9780521530149

Written to check the contents of the Cambridge syllabus. facts 2 corresponds to unit S2. It covers the Poisson distribution, linear mixtures of random variables, non-stop random variables, sampling and estimation, and speculation exams.

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Extra info for Advanced Level Mathematics: Statistics 2

Sample text

E. the secondstage problem, in the objective function, we can also write the above problem in a slightly different way: min cT x + qT y¯ : y¯ ∈ arg min qT y : Wy = z(ω) − T x, y ∈ Y , x ∈ X . 6). 5) can be seen as a two-stage optimization problem with the following counterparts: x = O, y = u, cT x = α qT y = O O 1 dx + β ∂O 1 ds, j (x, u(O; ω)) dx + ∂O k (x, u(O; ω)) ds. So the ﬁrst-stage decision in the case of the random shape optimization problem is the decision on the shape O. This decision has to be taken without knowing the actual forces, in other words non-anticipatively.

For t˜ ∈ Rl+1 deﬁne ⎧⎛ ⎞T ⎛ ⎞ ⎪ y ⎨ 0 ˜ Φ(t˜) := min ⎝1⎠ ⎝ v ⎠ : ⎪ ⎩ 0 w W qT = min v : Wy = t˜1 , y ≥ 0, ⎫ ⎛ ⎞ ⎪ y ⎬ 0 0 ⎝ ⎠ v = t˜, v, w ≥ 0, y ≥ 0 ⎪ −1 1 ⎭ w qT y − t˜2 ≤ v, 0 ≤ v . Then we have that ˜ ω) := 0 G(x, 0 T η ˜ +Φ x z(ω) 0 − 0 −1 T cT η x 24 1 Introduction ˜ =Φ z(ω) − T x η − cT x = min v : Wy = z(ω) − T x, y ≥ 0, qT y + cT x − η ≤ v, 0 ≤ v ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ = max cT x − η + min qT y : Wy = z(ω) − T x, y ≥ 0 , 0 ⎪ ⎪ ⎩ ⎭ =Φ(z(ω)−T x) = max {G(x, ω) − η, 0} . Since we restricted ourselves to the ﬁnite scenario case, this yields without further ˜ ω)).

For t˜ ∈ Rl+1 deﬁne ⎧⎛ ⎞T ⎛ ⎞ ⎪ y ⎨ 0 ˜ Φ(t˜) := min ⎝1⎠ ⎝ v ⎠ : ⎪ ⎩ 0 w W qT = min v : Wy = t˜1 , y ≥ 0, ⎫ ⎛ ⎞ ⎪ y ⎬ 0 0 ⎝ ⎠ v = t˜, v, w ≥ 0, y ≥ 0 ⎪ −1 1 ⎭ w qT y − t˜2 ≤ v, 0 ≤ v . Then we have that ˜ ω) := 0 G(x, 0 T η ˜ +Φ x z(ω) 0 − 0 −1 T cT η x 24 1 Introduction ˜ =Φ z(ω) − T x η − cT x = min v : Wy = z(ω) − T x, y ≥ 0, qT y + cT x − η ≤ v, 0 ≤ v ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ = max cT x − η + min qT y : Wy = z(ω) − T x, y ≥ 0 , 0 ⎪ ⎪ ⎩ ⎭ =Φ(z(ω)−T x) = max {G(x, ω) − η, 0} . Since we restricted ourselves to the ﬁnite scenario case, this yields without further ˜ ω)).