equal
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80 |
80 |
81 :def:`Variance` is a: |
81 :def:`Variance` is a: |
82 |
82 |
83 .. math:: |
83 .. math:: |
84 |
84 |
85 var[X] = E[(X - E[X])^2] = E[X^2] - E^2[X] |
85 var[X] = E[(X - E[X])²] = E[X²] - (E[X])² |
86 |
86 |
87 :def:`Standard deviation` is a: |
87 :def:`Standard deviation` is a: |
88 |
88 |
89 .. math:: |
89 .. math:: |
90 |
90 |
103 Let :math:`A_i ∩ A_j = ∅` for :math:`i ≠ j` and :math:`∑_i\ A_i = Ω`: |
103 Let :math:`A_i ∩ A_j = ∅` for :math:`i ≠ j` and :math:`∑_i\ A_i = Ω`: |
104 |
104 |
105 .. math:: |
105 .. math:: |
106 |
106 |
107 p_X(x) = Σ_i P(A_i)·p_{X|A_i}(x) |
107 p_X(x) = Σ_i P(A_i)·p_{X|A_i}(x) |
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108 |
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109 * https://en.wikipedia.org/wiki/Law_of_total_probability |
108 |
110 |
109 Conditional PMF on event |
111 Conditional PMF on event |
110 ======================== |
112 ======================== |
111 |
113 |
112 :def:`Conditional PMF on event` is: |
114 :def:`Conditional PMF on event` is: |
494 negative; |
496 negative; |
495 x |
497 x |
496 ------------ |
498 ------------ |
497 2 |
499 2 |
498 x - 2 x + 1 |
500 x - 2 x + 1 |
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501 |
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502 .. math:: |
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503 |
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504 E[(geom(p))²] = ∑_{x=1..∞} x²·p·(1-p)^(x-1) |
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505 |
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506 = p·∑_{x=1..∞} x²·(1-p)^(x-1) |
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507 |
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508 = p/(1-p)·∑_{x=0..∞} x²·(1-p)^x |
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509 |
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510 = p/(1-p)·(1-p)·(1-p+1)/(1 - (1-p))³ = p·(2-p)/p³ = (2-p)/p² |
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511 |
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512 .. NOTE:: |
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513 |
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514 Maxima calculation:: |
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515 |
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516 load("simplify_sum"); |
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517 (%i3) assume(x>0); |
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518 (%o3) [x > 0] |
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519 (%i4) assume(x<1); |
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520 (%o4) [x < 1] |
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521 |
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522 (%i8) simplify_sum(sum(k^2 * x^k, k, 0, inf)); |
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523 2 |
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524 x + x |
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525 (%o8) - ------------------- |
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526 3 2 |
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527 x - 3 x + 3 x - 1 |
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528 |
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529 So: |
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530 |
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531 .. math:: var(geom(p)) = E[(geom(p))²] - E[geom(p)]² = (2-p)/p² - 1/p² = (1-p)/p² |