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1 Basic Probability laws 0 ≤ P (X = x) ≤ 1 (1) where X is the random variable, and x is the value of the random variable. n X P (X = xi ) = 1 (2) i=1 where x1 , x2 , · · · , xn are mutually exclusive. Mutually exclusive: One event precludes the occurance of another event. Ex: Toss of a coin in which both H and T cannot occur at the same time in one trial. So H and T are mutually exclusive events. If x1 , x2 , · · · , xn are mutually exclusive then P (X = x1 ∪ X = x2 ∪ · · · ..... ∪ X = xn ) = P (X = x1 ) ∪ P (X = x2 ) ∪ · · · ..... ∪ P (X = xn ) (3) Probability of an event in set A or in set B to occur is given by P (A ∪ B) = P (A) + P (B) − P (A ∩ B) (4) if A and B have something in common (not mutually exclusive). Otherwise it is P (A ∪ B) = P (A) + P (B) (5) if A and B have nothing in common (mutually exclusive). Independent events: If A and B are independent trials, that is both trials can occur and are independent of each other, then P (A ∩ B) = P (A) × P (B) (6) Ex: Tossing a coin 2 times and getting Head twice. Here the event of getting H occur in independent trials. The probability of getting two heads is 0.5 × 0.5, which is 0.25. 2 Random variable X X takes on values xi . Probability of xi is P (X = xi ) or P (xi ). 2.1 Discrete Random variable pmf= probability mass function p(xi ) = P (X = xi ), cdf = cumulative distribution function F (xi ) = P P (X ≤ xi ) = ii=0 p(xi ) 2.2 Continuous Random variable pdf= probability density function f (x), cdf = cumulative density function F (x) = 2.3 Rb a f (x)dx Expected Value of a Random variable E(X) = n X xi p(xi ) (7) xf (x)dx (8) i=0 Z n E(X) = 0 1 2.4 Variance of a Random variable V ar(X) = E(X 2 ) − (E(X))2 (9) E(X 2 ) = E(X 2 ) = n X (xi )2 p(xi ) i=0 Z n x2 f (x)dx (10) (11) 0 3 3.1 Discrete Distributions Binomial x successes in n trials P (X = x) = (nCx)px (1 − p)n−x (12) where p is the probability of success (Ex. success could be defined as finding a defect in manufacturing). The mean is np and the variance is np(1 − p). Please note: the distribution is for the number of successes. 3.2 Geometric First success in n trials P (X = n) = p(1 − p)n−1 (13) Please note: the distribution is for the number of trials. The mean is (1/p) and the variance is (1 − p)/p2 . 3.3 Negative Binomial xth success in the nth trial P (X = n − x) = {(n − 1)C(x − 1)}px (1 − p)n−x (14) This means there are x − 1 successes in n − 1 trials. There are a total of n − x failures. The mean is (x/p) and the variance is x(1 − p)/p2 . Negative Binomial is also a sum of Geometric distributions. Please note: the distribution is for the number of failures. 3.4 Hypergeometric D defects in N , x defects in n, where n is sampled from N . P (X = x) = {(N − D)C(n − x)}{(D)C(x)} {(N )C(n)} 2 (15) 3.5 Poisson n is large and p is very small. Ex: Arrival process. λ = np. P (X = x) = e−λ λx x! (16) mean = variance = λ 3.6 4 Poisson approximation to Binomial Continuous Distributions 4.1 Exponential f (x) = λe−λx Z x F (x) = f (x)dx = 1 − e−λx (17) (18) 0 mean = 1/λ, variance = 1/λ2 4.2 Normal mean = µ, std. dev = σ, Normal = N(µ, σ), Standard Normal = Z(0,1) Z = (X − µ)/σ 5 (19) Conditional Probability Probability of event A given event B has already occurred is P (A|B) = P (A and B) P (B) (20) P (B|A) = P (A and B) P (A) (21) P (A|B) = P (B|A)P (A) . P (B) (22) Similarly, Therefore, combining the above P (B) = P (B|A)P (A) + P (B|Ā)P (Ā) (23) where Ā is the complement of A or in other words (not in A) = Ā. 5.1 Bayes Theorem Combining the above two equations (9 and 10) yields the Bayes Theorem P (A|B) = P (B|A)P (A) . P (B|A)P (A) + P (B|Ā)P (Ā) 3 (24)