Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
| | Joint probability mass function: , 0 (1) (2) ∑ ∑ , 1 (3) , , , 1 Marginal probability mass function: | | 1 0 – Mutually exclusive 0 – Unbiased Two events are independent if: | (1) | (2) (3) Probability density function: (1) 0 (2) 1 Discrete uniform distribution, all n has equal probability: 1 1 1 , , 2 12 Continuous uniform distribution: 1 , , 12 2 A Bernoulli trial (Binomial distribution, success or failure) 5 1 1 5 1 / min 1 1 0, 1, 2, … , 5 Cumulative distribution function: , ∞ ∞ Expected value of a function of a continuous random value: Normal distribution: 1 , , √2 Standardizing to calculate a probability: Normal approximation to the binomial distribution: 1 is approximately a standard normal rv. To approximate a binomial probability with normal distribution 0.5 0.5 1 and 0.5 0.5 1 The approximation is good for 5 and 1 5 Normal approximation to the Poisson distribution: , 5 √ Exponential distribution: λe , for 0 x ∞ 1/ , 1/ | | is a probability mass | | | For discrete random variable X & Y, if one of the following is true, the other are also true and X, Y are independent: (1) , , & , & 0 (2) | (3) , & 0 | (4) , Covarience: ∑ , , , , ! Permutations: 1 1 0 ! ! ! The standard error of an estimator Θ is its standard deviation, given by: Θ The mean squared error of the estimator Θ of the parameter θ is: Θ Θ θ ∏ Likelihood function: Suppose a random experiment consists of a series of n trials. Assume that 1) The result of each trial is classified into one of k classes. 2) The probability of a trial generating a result in class 1 to class k is constant over trials and equal to p1 to pk. 3) The trials are independent The random variables X1, X2,…,Xn that denote the number of trials that result in class 1 to class k have a multinomial distribution and the joint probability mass function is ! , ,…, … ! !… ! 1 , 1 If x1, x2, … , xn is a sample of n observations, the sample variance is: ∑ ∑ ∑ 1 1 Confidence interval on the mean, variance known: /√ /√ / / / | | , , /√ Confidence interval on the mean, variance unknown: /√ /√ / , / , /√ Random sample normal disr. mean=µ, var=σ2, S2=sample var. 1 . 1 Ci on variance, s2=sample variance, σ2 unknown 1 1 , , Lower and upper confidence bounds on σ2: 1 1 , | If X & Y are independent random variables, ! Combinations: max 0, , , | 1,2, … Poisson distribution: ! | Correlation: Negative binomial distribution nr of trials until r successes: 1 1 , 1, 2, … , , 1, 2, … 1 1 / / , Hypergeometric distribution: N objects contains K objects = successes, N‐K objects = failures. A sample of size n, X = # of successes in the sample. p=K/N, (good for n/N < 0.1) , , Because a conditional probability mass function function, the following prop. are satisfied: 0 (1) | (2) ∑ | 1 | (3) | Conditional mean, variance: 1 Conditional probability mass function of Y given X=x is: , , 0 | (3) Probability mass function: (1) 0 (2) ∑ 1 (3) Mean and variance of the discrete rv: , Geometric distribution: 1 , 1 , , , Binomial proportion: If n is large, the distribution of Approximate test on a binomial proportion: : 1 1 ̂ 1 ̂ ̂ 1 ̂ / ̂ / 1 Alternative hypothesis Rejection criteria (**) H1: p ≠ p0 z0 > zα/2 or z0 < ‐zα/2 H1: p > p0 z0 > zα z0 < ‐zα H1: p < p0 App. Sample size for a 2‐sided test on a binomial proportion: is approximately standard normal. Ci on binomial proportion (obs, lower, upper change zα/2 to zα): ̂ , 1 / 1 , 1 Sample size for a specified error on binomial proportion: 1 , max Test on the differens in mean, variance known ∆ : ∆ , 0.5 Prediction int. on a single future observation from norm. dist: 1 1/ 1 1/ / , / , Ci, difference in mean, variances known: / Alternative hypothesis Rejection criteria H1: ∆ z0 > zα/2 or z0 < ‐zα/2 H1: ∆ z0 > zα H1: ∆ z0 < ‐zα Sample size, 1‐sided test on difference in mean, with power of at least 1‐β, n1=n2=n, variance known: / for one‐sided, change zα/2 to zα. Sample size for a c.i. on difference in mean, variances known: / ∆ ∆ Tests on diff. in mean, variances unknown and equal: ∆ : ∆ , 1 1 Ci Case 1, difference in mean, variance unknown & equal: 1 1 / , 1 1 / , Alternative hypothesis Rejection criteria ∆ H1: / , / , H1: ∆ , ∆ H1: , Tests on diff. in mean, variances unknown and not equal: ∆ If : ∆ is true, the statistic 1 1 2 Ci Case 2, difference in mean, variance unknown, not equal: / , / , / 1 1 ν is degrees of freedom for tα/2,if not integer, round down. Ci for µD from paired samples: /√ /√ / , / , Approximate ci on difference in population proportions: / ̂ ̂ 1 ̂ ̂ ̂ 1 is distributed app. as t with ν degrees of freedom, ~ Paired t‐test: : ̂ ̂ 1 ̂ ̂ 1 ̂ / : , /√ , 1 1 , , Goodness of fit: , , Expected frequency: The power of a test: 1 Φ z Δ Rejection criteria z0 > zα/2 or z0 < ‐zα/2 z0 > zα z0 < ‐zα Δ , 2 Δ σ n ∑ , ∑ , Ci: , / , / , , , 1 Δ / σ n ∑ ∑ , z σ n ∑ Rejection criteria / , Φ The P‐value is the smallest level of significance that would lead to rejection of the null hypothesis H0 with the given data. 21 Φ | | : : : 1 Φ z : : : Φ z : : : Fitted or estimated regression line: 1 1 , , , / , / σ n Alternative hypothesis Rejection criteria H1: µ ≠ µ0 t0 > tα/2,n‐1 or t0 < ‐tα/2, n‐1 H1: µ > µ0 t0 > tα,n‐1 H1: µ < µ0 t0 < ‐tα,n‐1 Test in the variance of a normal distribution: 1 : , Alternative hypothesis H1: σ2 ≠ H1: σ2 > H1: σ2 < : 1 Hypothesis test: 1. Choose parameter of interest 2. H0: 3. H1: 4. α= 5. The test statistic is 6. Reject H0 at α= … if 7. Computations 8. Conclusions Test on mean, variance known : , /√ Alternative hypothesis H1: µ ≠ µ0 H1: µ > µ0 H1: µ < µ0 Test on mean, variance unknown , /√ Alternative hypothesis Rejection criteria H1: ≠ Δ t0 > tα/2,n‐1 or t0 < ‐tα/2, n‐1 t0 > tα,n‐1 H1: > Δ H1: < Δ t0 < ‐tα,n‐1 Approximate tests on the difference of two population proportions: ̂ / ̂ Δ , , ,