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RANDOM VARIABLES – DISCRETE- (2)
CONTINOUS RANDOM VARIABLE (4)
TOTAL PROBABILITY THEOREM (5)
RELATIONSHIP BETWEEN EXPECTED VALUE AND VARIANCE (6)
POINT ESTIMATION (7)
MARKOV INEQUALITY – CHEBICHEV INEQUALITY (8)
VECTOR RANDOM V, JOINT P DENSITY (9)
INDIPENDENCE, EXPECTED VALUE AND COVARIANCE CORRELATION (10)
CAUCHY-SCHWARZ INEQUALITY (11)
PROPERTIES OF THE VARIANCE (12)
MOMENT GENERATING FUNCTION (12)
CENTRAL LIMIT THEOREM (13)
WAYS TO FIND ESTIMATOR: METHOD OF MOMENT AND MAXIMUM LIKELIHOOD (14)
RAO CRAMER INEQUALITY (14)
SUFFICIENT ESTIMATOR (15)
INTERVAL OF CONFIDENCE (16)
HYPOTESIS TEST (16)
P-VALUE (17)
SIMPLE LIKELIHOOD RATIO (19)
LINEAR INTERPOLATION (21)
LINEAR STATISTICAL MODEL (22)
DIFFERENCES BETWEEN DISCRETE AND CONTINOUS RANDOM V.
A variable is a quantity whose value changes. A discrete variable is a variable whose value is obtained by counting. Examples : number of students present number of red marbles in a jar number of heads when flipping three coins students’ grade level A continuous variable is a variable whose value is obtained by measuring. Examples : height of students in class weight of students in class time it takes to get to school distance traveled between classes A random variable is a variable whose value is a numerical outcome of a random phenomenon. ▪ A random variable is denoted with a capital letter ▪ The probability distribution of a random variable X tells what the possible values of X are and how probabilities are assigned to those values ▪ A random variable can be discrete or continuous A discrete random variable X has a countable number of possible values. Discrete Random Variables:
- UNIFORM: is a symmetric probability distribution wherein a finite number of values are equally likely to be observed o Throw a regular dice. Every face, asymptotically, has the same p; o Used for generating random numbers.
- BERNOULLI: is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability 1 - p o Two outcomes, mutually exclusive. Example: Throw a coin; o Probability p of success is the same for every experiment; o All experiments are independent.
- BINOMIAL: A binomial random variable counts how often a particular event occurs in a fixed number of trials (independent) o A set of n (FINITE) Bernoulli’s experiments.
- HYPERGEOMETRIC: is a discrete probability distribution that describes the probability of successes (random draws for which the object drawn has a specified feature) in draws, without replacement, from a finite population o Card games with extraction and not reemission.
- GEOMETRIC: discrete probability distribution that describes the number of trial X needed to get one success o Two possible outcomes. o Independent trials (means that u don’t become savvier each trial);
Continuous Random Variables: is a random variable where the data can take infinitely many values.
- CONTINUOUS UNIFORM or RECTANGULAR DISTRIBUTION o Same concept of the DISCRETE UNIFORM but in the continuous case, then INFINITE support uniformly distributed within b and a.
- TRIANGULAR
- Triangularly shaped; o Features: § Minimum value; § Maximum value; § Mode (highest frequency value).
- EXPONENTIAL o Its probability density function has the highest value in zero and then all the way down to zero, therefore is suitable for all those experiments with monotonously decreasing probability like product reliability, how long a product lasts, the amount of time to deliver a service o Watch out for the DECAY FACTOR in the formula. Lambda is the reciprocal of the MEAN; o DIFFERENCE POISSON vs EXPONENTIAL. The poisson has a FIXED period of time, is discrete, defined in INTEGERS (infinite, but COUNTABLE anyway). The exp has a concept of continuous or infinite time, is a continuous random variable therefore infinite NOT COUNTABLE as well, NOT only integers.
- NORMAL
- Symmetrical on both sides of the mean
- Mean=Mode=Median;
- Tails are asymptotic, approach the x-axis but never meet;
- MAIN PARAMETERS: o Mean o Variance or Standard Deviation (squared root of the variance)
- Shape like a bell
- Most famous form: STANDARD NORMAL DISTRIBUTION
- Within 1 st dev from the mean (in excess or defect) ~68% p
- Within 2 st devs from the mean (in excess or defect) ~95.4% p
- Within 3 st devs from the mean (in excess or defect) ~99.7% p
- To use TABLES, you have to STANDARDIZE X into Z
- GAMMA o Used to measure the time between the occurrence of events when the event process is not completely random. Therefore, insurance risk theory, inventory control, amount of rainfall. o ASSUMPTION: independent occurrences. no memory;
o SUPPORT: x>0 ; o More general case of the EXPONENTIAL DISTRIBUTION (see formulae); o PARAMETERS: § ALPHA: SHAPE. Related to SKEWNESS of the distribution;
- if alpha=1 EXPONENTIAL DISTRIBUTION. § LAMBDA or BETA: SCALE. Related to KURTOSIS (allontanamento dalla normalità distributive) of the distribution.
- if lambda=1 STANDARD GAMMA DISTRIBUTION.
- CHI SQUARE with k degrees of freedom o Comes from a GAMMA with r=0.5 and lambda=k/ LINGO:
- PROBABILITY DENSITY FUNCTION (PDF) as funzione di densità
- CUMULATIVE DISTRIBUTION FUNCTION (CDF) as funzione di ripartizione discrete
- TOTAL PROBABILITY THEOREM: It expresses the total probability of an outcome which can be realized by several distinct events o with all B mutually exclusive o union of all B gives OMEGA LECTURE 3 (MARZOCCHI)
POINT ESTIMATION
• GOODNESS:
o CORRECT ESTIMATOR of Theta (parameter to estimate) § this impact the EFFICIENCY (MSE). then the BIAS is zero o DISTORTION or BIAS of T o EFFICIENCY (hat is the estimator. no hat is the parameter) o if n is FINITE: CORRECTNESS and EFFICIENCY (MSE) o if n is INFINITE: CONSITENCY (MSE) and ASYMPTOTIC CORRECTNESS § ASYMPTOTIC CORRECTNESS: § CONSITENCY + CORRECTNESS:
- Looking for pdf of FUNCTIONS OF A RANDOM VARIABLE:
zero!!! not
tetha
o CUMULATIVE DISTRIBUTION FUNCTION (preferred method) o CHANGE OF VARIABLE FORMULA § § This formula MAY FAIL IF g IS NOT STRICTLY INCREASING
- For example, in this case doesn’t work (slide 9/12)
- MARKOV INEQUALITY: gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant o support of X: only NULL or POSITIVE values. o from wikipedia it’s clearer:
- CHEBYSHEV INEQUALITY : for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean, only a definite fraction of values will be found within a specific distance from the mean of a distribution. o variance must be finite
• INDEPENDENCE
the same for CDF
- EXPECTED VALUE OF FUNCTIONS OF SEVERAL RV: integral of f(x) * x
- COVARIANCE o if X and Y are independent their COV= o the converse is not generally true (despite COV=0, 2 rv can still be dependent). example X is a uniform and Y is the module of X § but there’s one exception to this point is the 2 DIMENSIONAL NORMAL (see below) for which COV=0 if and only if X and Y are independent then if persists COV= there is independence as well.
- CORRELATION: it actually to the degree to which a pair of variables are linearly related
CAUCHY-SCHWARZ INEQUALITY: the square of the integral of the product of two functions is less than or equal to the product of the integrals of the square of each function. o Applied to the covariance proves that rho, in absolute terms, can be at most 1. this comes from the fact that the denominator of rho can be AT LEAST equal to the numerator… otherwise D is bigger than N
- 2D NORMAL DISTRIBUTION
- PROPERTIES OF THE SUM OF TWO VARIABLES numerator of rho (squared) denominator of rho (squared)
o if you multiply a constant a (-> 1/n) CONCLUSION: o you can retrieve other distributions from the MOMENT GENERATING FUNCTION of the SUM or the AVERAGE § SUM: n INDEPENDENT Bernoulli - > BINOMIAL § SUM: n INDEPENDENT Exponential - > GAMMA
- CENTRAL LIMIT THEOREM: states that the distribution of a sample variable approximates a normal distribution (i.e., a “bell curve”) as the sample size becomes larger. o X1,…,Xn EQUALLY DISTRIBUTED and INDEPENDENT o all with the same parameters mu and sigma square o For THE AVERAGE, with n to infinite EVERY TYPE of distribution tends to a STANDARD NORMAL with parameters mu and sigma_square/radq(n) § for THE SUM is equivalent except for the parameters which are nmu* and nsigma_square* o EXAMPLE: Bernoulli 12/ o THEOREM with strong assumptions which doesn’t hold if: § n small § p and q far from 0, o EXCEPTION TO CLT: SAMPLE VARIANCE o even if X1,…,Xn EQUALLY DISTRIBUTED and INDEPENDENT the CLT doesn’t hold for the SAMPLE VARIANCE o anyway,
WEEK 8 (MARZOCCHI) SLIDES
- ways to find ESTIMATORS: o METHOD OF MOMENTS : method of estimation of population parameters. You take known facts about the population and extend those ideas to a sample. It starts by expressing the population moments (i.e., the expected values of powers of the random variable under consideration) as functions of the parameters of interest. Those expressions are then set equal to the sample moments. § the solution of the system is the desired estimator o METHOD OF MAXIMUM LIKELIHOOD : a method of estimating the parameters of a probability distribution, that selects as estimates those parameter values maximizing the probability of obtaining the observed data § fix x1,…,xn find THE MAXIMUM tetha hat for L. - that will be the function of x1,…,xn called MAXIMUM LIKELIHOOD ESTIMATOR of tetha § INVARIANCE OF MAXIMUM LIKELIHOOD ESTIMATORS - suppose a monotonous function tau - EXAMPLE: if the estimator for variance is S^2, the estimator for dev std is radq(S^2)
- UNBIASED/UNDISTORTED/CORRECT ESTIMATORS o sample mean is UNBIASED o correct variance is BIASED. doesn’t equal sigma squared
- MEAN SQUARE ERROR (MSE) o var T o BIAS/DISTORTION (squared)
- RAO-CRAMER INEQUALITY o to get EFFICIENT estimators the MSE must be the lowest possible (excluding biasness or not, even though this theorem works for UNBIASED ESTIMATORS ). And this happen if THE VARIANCE (MSE component) IS MINIMAL. o this theorem sets a LOWER BOUND FOR THE VARIANCE of an UNBIASED ESTIMATOR o if an UNBIASED ESTIMATOR reaches the lower bound of lowest variance is said to be OPTIMAL or UMVUE (Uniformly Minimal variance Unbiased Estimator).
INTERVAL OF CONFIDENCE: a range of estimates for an unknown parameter, defined as an interval with a lower bound and an upper bound. is a range of values that’s likely to include a population value (such as the mean). with a certain degree of confidence. HYPOTHESIS TESTS (Practice) PARAMETRIC TEST o statistical distribution of the population known
- NON-PARAMETRIC TEST o otherwise
- TEST OF SIGNIFICANCE o measures the evidence with respect to the NULL HYPOTHESIS
- HYPOTHESIS TEST o contemplates the ALTERNATIVE HYP against H
- ACCEPTANCE REGION
- REJECTION REGION o the greater the SIGNIFICANCE ALPHA the greater the REJECTION REGION
- CRITICAL VALUES o values delimiting the REJECTION REGION
- HOW TO PERFORM THE TEST? if the condition is satisfied REJECT H0 AND ACCEPT H1!!!
- ERRORS o alpha= SIGNIFICANCE LEVEL of the test. PROB OF MAKING ERROR OF FIRST TYPE o beta= PROB OF MAKING AN ERR OF SECOND TYPE RIGHT TAIL!!! means that alpha must be at the RIGHT of the quantile Doesn't mean changing the sign of a possibly already negative quantile!!!
§ this last two have INVERSE RELATIONSHIP. § increasing the sample size n you can lower them down § TO CALCULATE THEM U MUST BE PROVIDED OF SOME INPUT ABOUT THE VALUE OF THE STATISTICS TEST o 1 - alpha=CONFIDENCE COEFFICIENT o 1 - beta=POWER OF THE TEST
- P VALUE o prob of REJECTING H0 while the HYPOTHESIS WAS TRUE indeed (ERROR OF FIRST TYPE?) o THE SMALLER THE P VALUE THE GREATER THE EVIDENCE AGAINST H0 (REJECT H0) o it’s a PROBABILITY. instead of a QUANTILE (like in hypothesis testing) o the PROB CALCULUS follows the table. LOOK ONLY AT THE SIGNS OF INEQUALITIES!!! o
- BIDIRECTIONAL TEST o unlike the DIRECTIONAL TEST. it checks BOTH DIRECTIONS
- HYPOTHESIS TEST: VARIANCE o so far it has been for the mean but know we want the VARIANCE o S^2 correct estimator of variance o sigma^2_0 variance in H o distributed like a CHI SQUARE with n-1 degrees of freedom RIGHT TAIL!!! means that alpha must be at the RIGHT of the quantile
o H1: no cancer § ERROR 1° TYPE: reject “got cancer” but it’s true § ERROR 2° TYPE: accept “got cancer” but it’s false.
- 1 - alpha= LEVEL OF SIGNIFICANCE. H0 true and you CORRECTLY chose H
- 1 - beta= POWER OF THE TEST. H0 false and you CORRECTLY chose H although on Marzocchi’s slide it says that POWER OF THE TEST is the probability of REJECTING H despite H0 is true or false. my idea is that there is the POWER OF TEST CORRECT (H0 false and you CORRECTLY chose H1) and INCORRECT (H0 true and you BADLY chose H1)-> and this last one is THE ERROR OF 1° TYPE (they are linked? holy shit) Even on the internet: Statistical power, or the power of a hypothesis test is the probability that the test correctly rejects the null hypothesis. Then, if they ask you what is: first draw the table of the 4 occurrences than point to the fact that is REJECTING H0 when H0 FALSE anyway, POWER OF THE TEST (probability of rejecting H0) must be: o LOW when H0 is true o HIGH when H0 is false
- AMPLITUDE OF A TEST o if H0 is a subset of all the possible values of the parameter o a is the amplitude. THE LARGEST PROBABILITY OF MAKING AN ERROR OF THE FIRST TYPE o 1 - a: CONFIDENCE COEFFICIENT. o THERE IS A TRADEOFF BETWEEN HAVING A HIGH CONFIDENC<e COEFFICIENT ( 1 - alpha .low amplitude) and ERROR OF 2° TYPE. if you increase a lot 1-a (decreasing the amplitude) you are increasing THE BOUNDARY TO SURPASS TO REJECT H0, then you are ACCEPTING H0 more and more HAVING MORE PROBABILITY THAT H0 WAS FALSE (ERR 2° TYPE). SOLUTION: take a large n.
- SIMPLE LIKELIHOOD RATIO : assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods. o the parameter can be “ONE TYPE or ANOTHER”. § H0: tetha=tetha § H1: tetha=tetha o Lk is the probability of getting the data x1,...,xn when tetha=tethak (k=0,1)
• MOST POWERFUL TESTS
- AIM: keeping the same amplitude (ERR 1° TYPE) reduce the ERR 2° TYPE
- CONDITIONS: o just fix the amplitude as a o prob of REJECT H0 (when H1 TRUE) is the HIGHEST among all tests with AMPLITUDE LESS THAN a. SMALLEST ERR 2° TYPE AMONG ALL TESTS WITH AMPLITUDE LESS THAN a. highest POWER OF THE TEST among all tests with amplitude less than a ??? o NEYMAN-PEARSON LEMMA. the test of simple likelihood IS THE MOST POWERFUL. but is restricted to
- GENERALIZED LIKELIHOOD RATIO : assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint.
- CHI SQUARE TEST o the quantity for n->infinite it’s distributed like a CHI SQUARE with m degrees of freedom o useful for testing the PROBABILITIES, like o APPLICATION: calculate the quantity, compare (in the case above Q>xa REJECT H0) it with the quantile of the chi square obtained through tables and level of significance given and then decide if ACCEPT/REJECT H0.
- P VALUE o H0 vs NOT(H0)
