Chapter 2, part 2. Petter Mostad

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1 Chapter 2, part 2 Petter Mostad

2 Parametrical families of probability distributions How can we solve the problem of learning about the population distribution from the sample? Usual procedure: Assume the population distribution has a particular parametric form, and estimate parameters from the sample.

3 Example: The normal distribution A family of probability distributions. Two parameters: μ and σ>0 The probability density is p( x) exp ( x µ ) σ = πσ For all values of the parameters, the integral is, so it is a probability distribution. The expectation is μ and the standard deviation is σ. 2

4 Plots of the normal distribution The standard normal distribution has expectation 0 and standard deviation

5 Taking means of large samples Polulation Means of samples size Means of samples size Means of samples size Start with some prob distribution Take samples of size 3, 5, 20, and look at their means The distribution of such means will approach normal distribution as sample size increases

6 The central limit effect Applies to almost any population distribution The distribution of means will tend to normal as sample size increases In practice: Observations that are sums of many similar independent randomly distributed variables tend to be normal: (Examples: length of a person, scores on tests, )

7 Normal probability plots Sample Quantiles Normal Q-Q Plot Theoretical Quantiles Question: could the sample reasonably be from a normal population? For other distributions: qqplots What can be leaned from such plots Using them for a rough estimate of the standard deviation

8 Statistical dependence The joint distribution of two random variables The conditional distribution of one given the other Independence / dependence IID, NIID

9 Covariance and correlation Sample covariance for data Sample correlation: Can be computed with integrals for infinite populations represented by probability distributions. Linear dependence is measured ) )( ( y y x x n i n i i = ), ),...,(, ),(, ( 2 2 n x n y y x y x = = = n i i n i i i n i i y y n x x n y y x x n 2 2 ) ( ) ( ) )( (

10 Autocorrelation For series of observations (e.g., time-series) The correlation between a sequence and its shifted version. lag k sample autocorrelation coefficient Is often positive; can be negative Will influence the accuracy of derived estimates!

11 Example: Autocorrelation No autocorrelation Some autocorrelation High autocorrelation Negative autocorrelation

12 Sampling distributions We previously found the best fitting normal distribution to model the population for a sample by computing sample mean and standard deviation, and using these as parameters in a normal distribution. This does not tell us how likely it is that other normal distributions with similar parameters could represent the population. To answer such questions, our textbook follows this path: Regard parameters as unknowns Find the sampling distributions of relevant summary statistics as functions of the parameters Learn about the unknown parameters from the sampling distributions and the value of the summary statistics.

13 Example: the mean of a sample from a normal distribution Population distribution Distrib. for mean of sample size Distrib. for mean of sample size Distrib. for mean of sample size Assume x,x 2,,x n is a sample from a norm. distr. with exp. μ and var. σ 2 Then x has a norm. distr. with exp. μ and var. σ 2 /n. For any distr. with exp. μ and var. σ 2, the sample average will have exp. μ and var. σ 2 /n. For large samples, the sample average will always have approx. normal distrib.

14 Example: the variance of a sample from a normal distribution Population distribution Distrib. for var. of sample size Distrib. for var. of sample size Distrib. for var. of sample size Assume x,x 2,,x n is a sample from a norm. distr. with exp. μ and var. σ 2 2 Then ( n ) s divided by σ 2 has a χ 2 distribution with n- degrees of freedom. The sampling distribution has expectation σ 2

15 The χ 2 distribution Has one parameter, the degrees of freedom Expectation is equal to the degrees of freedom Chi-square with 2 d.f Chi-square with 6 d.f Chi-square with 4 d.f Chi-square with 2 d.f

16 Yet another example The population distribution Distribution with sample size Distribution with sample size Distribution with sample size Assume x,x 2,,x n is a sample from a norm. distr. with exp. μ and var. σ 2 2 Then ( x µ ) s / n has a t-distribution with n- degrees of freedom. Note: the distribution of the statistic is indepenent of μ and σ 2

17 Properties of (Student s) t distribution Has one parameter: the degrees of freedom. Has expectation zero; symmetric around zero. Is similar to the normal distribution but has longer tails When degrees of freedom increase, approaches normal distribution. Important in applications. Tables of quantiles!

18 The F distribution If X has χ 2 distr. with n d.f. and Y has χ 2 distr. with m d.f., then (X/n)/ (Y/m) has F distr. with n and m degrees of freedom. Important distribution for comparing variances F distrbution with 2 and df F distrbution with 00 and df F distrbution with 5 and 2 df F distrbution with 00 and 00 df

19 The Binomial distribution A discrete distribution Gives the probability of k successes in n independent trials, when probability of success in each trial is p. Parameters: n and p. Expectation: np. Variance: np(-p) Probability formula: p( y) n y n! y!( n y n y y n y = p ( p) = p ( p) y)! y! = y Approaches normality when n is large (n>5) and p is not close to 0 or (i.e., probability can then be approximated with a normal probability density with same expectation and variance).

20 The Binomial distribution Binomial with n = 5, p = 0.2 Binomial with n = 0, p = Binomial with n = 0, p = 0.4 Binomial with n = 30, p =

21 Example If each wine bottle has a 5% chance of tasting of cork, what is the probability that 2 of 0 bottles taste of cork? Answer (assuming independence): 0! !8! = 0.075

22 The Poisson distribution Used for the probability of the number of independent events happening in a unit time interval, when the rate of events per time interval is η. Example: If rate of accidents is 4 per year, then the actual number of accidents in a year is Poisson-distributed (assuming independence). Corresponds to the Binomial distribution with n p=η when n goes to infinity an p to zero. Approximately normal for large η. Probability function: p( y) = e η y η y!

23 Poisson distribution Poisson with rate.8 Binomial approximating Poisson Poisson rate 4.3; normal approx Poisson rate 20; normal approx

24 Example If the rate of a type of cancers in a city is 0 per year, what is the probability of observing zero or? e p(0) = e p() = ! 0! 0 = e 0 = 0e = =

25 Expectation and variance as operators on random variables When a population is represented as a random variable X with probability density p(x), its expectation E(X) can be computed as an integral: Its variance V(X) can be computed in terms of expectations: 2 V ( X ) = E ( X E( X )) E ( X ) = xp( x) dx ( )

26 Mean and variance of linear combinations of random variables We can prove mathematically: E ( ax + bx 2) = ae( X) + be( X 2) 2 2 V ( ax + bx 2) = a V ( X) + b V ( X 2) + 2ab Cov( X, X 2) where we define (( X E( X ))( X E( ))) Cov( X, X 2) = E 2 X 2

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