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Page 1 PROBABILITY AND STATISTICS Types of events ? Complementary events ? ? ? ? ? ? c E s E ?? The complement of an event E is set of all outcomes not in E. ? Mutually Exclusive Events Two events E & F are mutually exclusive iff P(E n F) = 0. ? Collectively exhaustive events Two events E & F are collectively exhaustive iff (E U F) = S Where S is sample space. ? Independent events If E & F are two independent events P(E n F) = P (E) * P(F) De Morgan’s Law ? ? ? ?? ?? ?? C C i n n i i1 i1 E = E U ? ?? ?? ?? ?? ?? C C i nn i i 1 i 1 E = E Axioms of Probability n 12 E ,E ,...........,E are possible events & S is the sample space. a. 0 = P (E) = 1 b. P(S) = 1 c. ? ? n n ii i=1 i1 P E = P E ? ?? ?? ?? ? for mutually exclusive events Page 2 PROBABILITY AND STATISTICS Types of events ? Complementary events ? ? ? ? ? ? c E s E ?? The complement of an event E is set of all outcomes not in E. ? Mutually Exclusive Events Two events E & F are mutually exclusive iff P(E n F) = 0. ? Collectively exhaustive events Two events E & F are collectively exhaustive iff (E U F) = S Where S is sample space. ? Independent events If E & F are two independent events P(E n F) = P (E) * P(F) De Morgan’s Law ? ? ? ?? ?? ?? C C i n n i i1 i1 E = E U ? ?? ?? ?? ?? ?? C C i nn i i 1 i 1 E = E Axioms of Probability n 12 E ,E ,...........,E are possible events & S is the sample space. a. 0 = P (E) = 1 b. P(S) = 1 c. ? ? n n ii i=1 i1 P E = P E ? ?? ?? ?? ? for mutually exclusive events Some important rules of probability P(A U B) = P(A) + P(B) – P(A ?B) P(A ? B) = P(A)* P ? ? B | A = P(B) * P ? ? A | B P ? ? A | B is conditional probability of A given B. If A & B are independent events P(A ?B) = P(A) * P(B) P(A | B) = P(A) P(B | A) = P(B) Total Probability Theorem P(A ?B) = P (A ?E) + P (B ?E) = P(A) * P(E |A) + P(B) * P(E |B) Baye’s Theorem P(A |E) = P(A ? E) + P (B ?E) = P(A)* P(E | A) + P(B) * P(E | B) Statistics ? Arithmetic Mean of Raw Data x x n ? ? x = arithmetic mean; x = value of observation ; n = number of observations ? Arithmetic Mean of grouped data ? ? ? ? ? fx x f ; f = frequency of each observation ? Median of Raw data Arrange all the observations in ascending order n 12 x x ............ x ? ? ? If n is odd, median = ? ? n1 2 ? th value If n is even, Median = ? ? ? ? th th nn value + 1 value 22 2 ? Page 3 PROBABILITY AND STATISTICS Types of events ? Complementary events ? ? ? ? ? ? c E s E ?? The complement of an event E is set of all outcomes not in E. ? Mutually Exclusive Events Two events E & F are mutually exclusive iff P(E n F) = 0. ? Collectively exhaustive events Two events E & F are collectively exhaustive iff (E U F) = S Where S is sample space. ? Independent events If E & F are two independent events P(E n F) = P (E) * P(F) De Morgan’s Law ? ? ? ?? ?? ?? C C i n n i i1 i1 E = E U ? ?? ?? ?? ?? ?? C C i nn i i 1 i 1 E = E Axioms of Probability n 12 E ,E ,...........,E are possible events & S is the sample space. a. 0 = P (E) = 1 b. P(S) = 1 c. ? ? n n ii i=1 i1 P E = P E ? ?? ?? ?? ? for mutually exclusive events Some important rules of probability P(A U B) = P(A) + P(B) – P(A ?B) P(A ? B) = P(A)* P ? ? B | A = P(B) * P ? ? A | B P ? ? A | B is conditional probability of A given B. If A & B are independent events P(A ?B) = P(A) * P(B) P(A | B) = P(A) P(B | A) = P(B) Total Probability Theorem P(A ?B) = P (A ?E) + P (B ?E) = P(A) * P(E |A) + P(B) * P(E |B) Baye’s Theorem P(A |E) = P(A ? E) + P (B ?E) = P(A)* P(E | A) + P(B) * P(E | B) Statistics ? Arithmetic Mean of Raw Data x x n ? ? x = arithmetic mean; x = value of observation ; n = number of observations ? Arithmetic Mean of grouped data ? ? ? ? ? fx x f ; f = frequency of each observation ? Median of Raw data Arrange all the observations in ascending order n 12 x x ............ x ? ? ? If n is odd, median = ? ? n1 2 ? th value If n is even, Median = ? ? ? ? th th nn value + 1 value 22 2 ? ? Mode of Raw data Most frequently occurring observation in the data. ? Standard Deviation of Raw Data ? ? ? ?? ?? ii 2 2 2 n x x n n = number of observations variance = 2 ? ? Standard deviation of grouped data ? ? ? ?? ?? 2 i i i i 2 2 N f x f x N fi = frequency of each observation N = number of observations. variance = 2 ? ? Coefficient of variation = CV = ? ? ? Properties of discrete distributions ? ? ? ? a. P x 1 ? ? ? ? ? ? b. E X x P x ? ? ? ? ? ? ? ? ?? 2 2 c. V x E x E x ? Properties of continuous distributions ? ? ? ? ?? ? ? f x dx 1 ? ? ? ? ? ?? ? ? x F x f x dx = cumulative distribution ? ? ? ? ? ?? ? ? ? E x xf x dx = expected value of x ? ? ? ? ? ? ? ?? ?? ?? 2 2 V x E x E x = variance of x Page 4 PROBABILITY AND STATISTICS Types of events ? Complementary events ? ? ? ? ? ? c E s E ?? The complement of an event E is set of all outcomes not in E. ? Mutually Exclusive Events Two events E & F are mutually exclusive iff P(E n F) = 0. ? Collectively exhaustive events Two events E & F are collectively exhaustive iff (E U F) = S Where S is sample space. ? Independent events If E & F are two independent events P(E n F) = P (E) * P(F) De Morgan’s Law ? ? ? ?? ?? ?? C C i n n i i1 i1 E = E U ? ?? ?? ?? ?? ?? C C i nn i i 1 i 1 E = E Axioms of Probability n 12 E ,E ,...........,E are possible events & S is the sample space. a. 0 = P (E) = 1 b. P(S) = 1 c. ? ? n n ii i=1 i1 P E = P E ? ?? ?? ?? ? for mutually exclusive events Some important rules of probability P(A U B) = P(A) + P(B) – P(A ?B) P(A ? B) = P(A)* P ? ? B | A = P(B) * P ? ? A | B P ? ? A | B is conditional probability of A given B. If A & B are independent events P(A ?B) = P(A) * P(B) P(A | B) = P(A) P(B | A) = P(B) Total Probability Theorem P(A ?B) = P (A ?E) + P (B ?E) = P(A) * P(E |A) + P(B) * P(E |B) Baye’s Theorem P(A |E) = P(A ? E) + P (B ?E) = P(A)* P(E | A) + P(B) * P(E | B) Statistics ? Arithmetic Mean of Raw Data x x n ? ? x = arithmetic mean; x = value of observation ; n = number of observations ? Arithmetic Mean of grouped data ? ? ? ? ? fx x f ; f = frequency of each observation ? Median of Raw data Arrange all the observations in ascending order n 12 x x ............ x ? ? ? If n is odd, median = ? ? n1 2 ? th value If n is even, Median = ? ? ? ? th th nn value + 1 value 22 2 ? ? Mode of Raw data Most frequently occurring observation in the data. ? Standard Deviation of Raw Data ? ? ? ?? ?? ii 2 2 2 n x x n n = number of observations variance = 2 ? ? Standard deviation of grouped data ? ? ? ?? ?? 2 i i i i 2 2 N f x f x N fi = frequency of each observation N = number of observations. variance = 2 ? ? Coefficient of variation = CV = ? ? ? Properties of discrete distributions ? ? ? ? a. P x 1 ? ? ? ? ? ? b. E X x P x ? ? ? ? ? ? ? ? ?? 2 2 c. V x E x E x ? Properties of continuous distributions ? ? ? ? ?? ? ? f x dx 1 ? ? ? ? ? ?? ? ? x F x f x dx = cumulative distribution ? ? ? ? ? ?? ? ? ? E x xf x dx = expected value of x ? ? ? ? ? ? ? ?? ?? ?? 2 2 V x E x E x = variance of x ? Properties Expectation & Variance E(ax + b) = a E(x) + b V(ax + b) = a 2 V(x) ? ? ? ? ? ? ? ? ? 1 2 1 2 E ax bx aE x bE x ? ? ? ? ? ? 22 1 2 1 2 V ax bx a V x b V x ? ? ? cov (x, y) = E (x y) – E (x) E (y) Binomial Distribution no of trials = n Probability of success = P Probability of failure = (1 – P) ? ? ? ? nx nx x P X x C P 1 P ? ? ? ? Mean = E(X) = nP Variance = V[x] = nP(1 – P) Poisson Distribution A random variable x, having possible values 0,1, 2, 3,……., is poisson variable if ? ? x e P X x x! ?? ? ?? Mean = E(x) = ? Variance = V(x) = ? Continuous Distributions Uniform Distribution ? ? ? ?? ? ? ? ? ? ? 1 if a x b fx ba 0 otherwise Mean = E(x) = ba 2 ? Variance = V(x) = ? ? 2 ba 12 ? Page 5 PROBABILITY AND STATISTICS Types of events ? Complementary events ? ? ? ? ? ? c E s E ?? The complement of an event E is set of all outcomes not in E. ? Mutually Exclusive Events Two events E & F are mutually exclusive iff P(E n F) = 0. ? Collectively exhaustive events Two events E & F are collectively exhaustive iff (E U F) = S Where S is sample space. ? Independent events If E & F are two independent events P(E n F) = P (E) * P(F) De Morgan’s Law ? ? ? ?? ?? ?? C C i n n i i1 i1 E = E U ? ?? ?? ?? ?? ?? C C i nn i i 1 i 1 E = E Axioms of Probability n 12 E ,E ,...........,E are possible events & S is the sample space. a. 0 = P (E) = 1 b. P(S) = 1 c. ? ? n n ii i=1 i1 P E = P E ? ?? ?? ?? ? for mutually exclusive events Some important rules of probability P(A U B) = P(A) + P(B) – P(A ?B) P(A ? B) = P(A)* P ? ? B | A = P(B) * P ? ? A | B P ? ? A | B is conditional probability of A given B. If A & B are independent events P(A ?B) = P(A) * P(B) P(A | B) = P(A) P(B | A) = P(B) Total Probability Theorem P(A ?B) = P (A ?E) + P (B ?E) = P(A) * P(E |A) + P(B) * P(E |B) Baye’s Theorem P(A |E) = P(A ? E) + P (B ?E) = P(A)* P(E | A) + P(B) * P(E | B) Statistics ? Arithmetic Mean of Raw Data x x n ? ? x = arithmetic mean; x = value of observation ; n = number of observations ? Arithmetic Mean of grouped data ? ? ? ? ? fx x f ; f = frequency of each observation ? Median of Raw data Arrange all the observations in ascending order n 12 x x ............ x ? ? ? If n is odd, median = ? ? n1 2 ? th value If n is even, Median = ? ? ? ? th th nn value + 1 value 22 2 ? ? Mode of Raw data Most frequently occurring observation in the data. ? Standard Deviation of Raw Data ? ? ? ?? ?? ii 2 2 2 n x x n n = number of observations variance = 2 ? ? Standard deviation of grouped data ? ? ? ?? ?? 2 i i i i 2 2 N f x f x N fi = frequency of each observation N = number of observations. variance = 2 ? ? Coefficient of variation = CV = ? ? ? Properties of discrete distributions ? ? ? ? a. P x 1 ? ? ? ? ? ? b. E X x P x ? ? ? ? ? ? ? ? ?? 2 2 c. V x E x E x ? Properties of continuous distributions ? ? ? ? ?? ? ? f x dx 1 ? ? ? ? ? ?? ? ? x F x f x dx = cumulative distribution ? ? ? ? ? ?? ? ? ? E x xf x dx = expected value of x ? ? ? ? ? ? ? ?? ?? ?? 2 2 V x E x E x = variance of x ? Properties Expectation & Variance E(ax + b) = a E(x) + b V(ax + b) = a 2 V(x) ? ? ? ? ? ? ? ? ? 1 2 1 2 E ax bx aE x bE x ? ? ? ? ? ? 22 1 2 1 2 V ax bx a V x b V x ? ? ? cov (x, y) = E (x y) – E (x) E (y) Binomial Distribution no of trials = n Probability of success = P Probability of failure = (1 – P) ? ? ? ? nx nx x P X x C P 1 P ? ? ? ? Mean = E(X) = nP Variance = V[x] = nP(1 – P) Poisson Distribution A random variable x, having possible values 0,1, 2, 3,……., is poisson variable if ? ? x e P X x x! ?? ? ?? Mean = E(x) = ? Variance = V(x) = ? Continuous Distributions Uniform Distribution ? ? ? ?? ? ? ? ? ? ? 1 if a x b fx ba 0 otherwise Mean = E(x) = ba 2 ? Variance = V(x) = ? ? 2 ba 12 ? Exponential Distribution ? ? x e if x 0 fx 0 if x 0 ?? ? ? ? ? ? ? ? ? Mean = E(x) = 1 ? Variance = V(x) = 2 1 ? Normal Distribution ? ? ? ? 2 2 2 x 1 f x e p , x 2 2 ?? ? ? ? ?? ? ? ? ? ? ? ? ?? ? ?? ?? Means = E(x) = µ Variance = v(x) = 2 ? Coefficient of correlation ? ? ? ? ? ? ?? cov x, y var x var y x & y are linearly related, if ? = ± 1 x & y are un-correlated if ? = 0 Regression lines ? ? ? xx ? = ? ? ? xy b y y ? ? ? yy ? = ? ? ? yx b x x Where x & y are mean values of x & y respectively xy b = ? ? ? ? cov x, y var y ; yx b = ? ? ? ? cov x, y var x ?? xy yx bbRead More
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