By G. Matthewson
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Intuitively, if A and B are disjoint, then knowing that A occurs implies that B does not, which is some new information about B unless B is impossible in the first place. 4 General Definition Generally, we say that a collection of events {Ai , i ∈ I} are mutually independent if for any finite subcollection {i, j, . . , k} ⊂ I one has P (Ai ∩ Aj ∩ · · · ∩ Ak ) = P (Ai )P (Aj ) · · · P (Ak ). Subtility The definition seems innocuous, but one has to be a bit careful. 2. The sample space Ω = {1, 2, 3, 4} has four points that have a probability 1/4 each.
Then, for any real number b, X ≤ b if Z ≤ F (b), which occurs with probability F (b), as desired. The second method uses the pdf. Assume that X is continuous with pdf f (x) and that P (a < X < b) = 1 and f (x) ≤ c. Pick a point (X, Y ) uniformly in [a, b]×[0, c] (by generating two uniform random variables). , if Y ≤ f (X), then keep the value X; otherwise, repeat. Then, P (a < X < a + ) = P [A|B] where A = {(x, y)|a < x < a + , y < f (x)} = [a, a + ] × [0, f (a)]. Then, P [A|B] = P (A)/P (B) with P (A) = f (a) /[(ba)c] and P (B) = 1/[(ba)c].
Intuitively, if A and B are disjoint, then knowing that A occurs implies that B does not, which is some new information about B unless B is impossible in the first place. 4 General Definition Generally, we say that a collection of events {Ai , i ∈ I} are mutually independent if for any finite subcollection {i, j, . . , k} ⊂ I one has P (Ai ∩ Aj ∩ · · · ∩ Ak ) = P (Ai )P (Aj ) · · · P (Ak ). Subtility The definition seems innocuous, but one has to be a bit careful. 2. The sample space Ω = {1, 2, 3, 4} has four points that have a probability 1/4 each.
Constructing an astronomical telescope by G. Matthewson
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