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| e^(-y/40) = 0.5 can be rewritten as what? |
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Let k >= 1 in any measurement,
at least 1-(1/k^2) data fall w/in k deviations of mean |
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| (n+1)C(k) = (n)C(k) + (n)C(k-1) |
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| 3 Axioms of Probability Model |
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1. P(A) >= 0
2. P(S) = 1
3. If A1, A2, A3,... is a sequence of pairwise mutuatlly disjoint events in S, then P(A1 U A2 U A3 U...) = Sum (infinity, i=1) P(Ai) |
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| Events A, B are independent if... |
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ANY one of the follow holds:
P(A|B) = P(A)
P(B|A) = P(B)
P(A intrsc B) = P(A)P(B) |
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| The Multiplicative Law of Probability |
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| P(A intsc B) = P(A)P(B|A) |
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| Setup the multinomial coefficient (9)C(3,3,3) |
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| Types of discrete distributions and when to use them |
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| If you want to know the # of successes, use BINOMIAL if the prob. is fixed or the pop. is large, and use HYPERGEOMETRIC if sampling w/o replacement. For the # of trials, use NEGATIVE BINOMIAL if 1+ success and GEOMETRIC if 1 success. |
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| Definition of a binomial experiment |
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| FIXED NUMBER of n INDEPENDENT event with FIXED PROBABILITY of a SUCCESS/FAILURE |
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1. lim as y -> -inf F(y)=0
2. lim as y -> inf F(y)=1
3. F(y) is non-decreasing |
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| Correction for continueity |
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If P(X=n) use P(n – 0.5 < X < n + 0.5)
If P(X>n) use P(X > n + 0.5)
If P(X≤n) use P(X < n + 0.5)
If P (XIf P(X ≥ n) use P(X > n – 0.5) |
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| a is the shape parameter and b is the scale parameter |
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[1*3*5*...*(2n+1)*sqrt(pi)] /
2^n |
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| Conditional probability function |
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| p(y1|y2) = p(y1,y2)/p2(y2) |
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| Conditional density function |
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| f(x|y) = f(x,y)/f2(y) aka the joint density function divided by the marginal density function |
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| Marginal density function |
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| If Y1 and Y2 are jointly continuous random variables, the marginal density function of x is f1(x) = integral(-inf,inf) f(x,y) dy |
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| (a^2)*E(Y1) + (b^2)*E(Y2) + 2abCov(Y1,Y2) |
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| 1/(n-1) Sum(n,i=1) (Yi - Ybar)^2 |
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| If R.V. Y has a pdf f(y) = {(3/7)y^2), 1 <= y <= 2; 0, elsewhere} THEN the E(Y) = integral(2,1) (3/7)y^3 dy |
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| Example of joint density function f(x,y) |
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light bulb, independent
f(x,y) = {1. (1/1,000,000)e^(-x/1000)e^(-y/1000), 0 <= x < inf, 0 <= y < inf;
2. 0, elsewhere} |
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| Marginal probability function |
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| If Y1 and Y2 are jointly discrete random variables, the marginal prob. function of Y1 is P(Y1=y) = p1(y) = Sum(y2) p(y,y2) |
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| Chi-square (n-1) distribution = |
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| P(A|B) = [P(A)P(B|A)]/[P(A)P(B|A)+P(notA)P(B|notA)] |
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| Variance is independent when... |
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