Lecture 2. Probability Distributions Theophanis Tsandilas

Transcription

1 Lecture 2 Probability Distributions Theophanis Tsandilas

2 Comment on measures of dispersion Why do common measures of dispersion (variance and standard deviation) use sums of squares: nx (x i ˆµ) 2 i=1 instead of sums of absolute residuals: nx i=1 x i ˆµ

3 Comment on measures of dispersion Working with absolute values can be difficult, but this is not the main reason. The measure of central tendency that minimizes the sums of absolute differences is the median, not the mean. And since the mean is the prevalent measure of central tendency, we commonly use sums of squares. However, for statistical methods that rely on medians, sums of absolute differences can be more appropriate.

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5 What is a probability distribution? Consider the population of all possible outcomes when throwing two dice. How probable is each sum S of counts from the two dice? The probability distribution provides the probability of occurrence of all possible outcomes in an experiment

6 Probability distribution of a population It is generally not known. However, we may have sufficient information about this distribution to meet the goals of our research. Statistical modeling: It is possible to rely on a small set of probability distributions that capture the key characteristics of a population. We can then infer the parameters of these model probability distributions from our sample. Why? We expect that the sample contains some information about the population from which it was sampled.

7 Example distributions Height: Women vs. Men

8 Example distributions US Income distribution (older than 2013)

9 Example distributions Half Marathon and Marathon race finish time Source:

10 Example distributions Distribution of most frequent English words

11 Discrete probability distributions The population (and hence its samples) contain discrete values, either finite or infinite in number e.g., {-3, 1, 0, 1, 2}, { blue, brown, green }, or {1, 2, 3,...} The probability of a value x in a population can be expressed as a function: f(x) = p(x = x) (the probability of random variable X taking value x) The function f(x) is called a probability mass function (pmf)

12 Discrete probability distributions This is a discrete probability distribution

13 Discrete probability distributions Probabilities should sum to 1: X f(x) =1 x2s The expected value (or mean): E(x) = X x2s f(x)x = =7 The mode is the most frequent value: which one? The median is the middle value: which one?

14 Symmetrical probability distributions When the mean coincides with the median The above is a symmetrical, unimode distribution

15 The binomial distribution Consider a population containing two values: 1 and 0 Let s set the probability of 1 to Pr(1) =.75 and the probability of 0 to Pr(0) =.25. A single sample (n = 1) from such a population is known as a Bernoulli trial. a coin flipping trial with a fair coin is a Bernoulli trial with Pr(Head) =.5 and Pr(Tail) =.5 If we perform n independent Bernoulli trials, their outcomes will follow a binomial distribution.

16 The binomial distribution If a random variable X has a binomial distribution, we write: X B(n, P ) where n and P are the parameters of the distribution: n is the number of Bernoulli trials P is the chance (probability) of success If we know n and P, we can fully describe the distribution

17 The binomial distribution Probability Mass Function (pmf) n f(x; n, P )= P x (1 P ) n x x The probability of exactly n - x failures The probability of exactly x successes The number of possible ways that n Bernoulli trails lead to x successes

18 The binomial distribution Consider the distribution of errors for 10 trials, when for each trial, errors occur with a probability of 40%

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20 The binomial distribution Cumulative Distribution Function (cdf)

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22 Continuous distributions Not restricted to specific values. They can take any value between the lower and upper bound of a population (of course, populations can be unbounded).

23 Continuous distributions The probability of any particular value is zero. Probabilities can only be obtained for intervals (i.e., a range of values): Z b Pr(a apple X apple b) = f(x)dx a where f(x) is the probability density function (pdf). It provides the relative (rather than absolute) likelihood that a random variable X takes the value x. Pr( 1 apple X apple1)= Z 1 1 f(x)dx =1

24 Continuous distributions Mode: value of highest peak Median: value that divides the area exactly in half Mean: µ = Z 1 1 xf(x)dx

25 The normal distribution Also known as the Gaussian distribution

26 The normal distribution Symmetrical, unimodal and continuous Can be derived as a sum of an infinite number of independent random variables Thus, it is appropriate when data arise from a process that involves adding together the contributions from a large number of independent, random events.

27 Example The human height can be considered as the outcome of many independent random genetic and environmental influences

28 Normal distribution parameters A normal distribution can be fully described by two only parameters: its mean μ and its variance σ 2 A normally distributed variable X can be written as X N(µ, 2 ) Its probability density function (pdf) is as follows: f(x; µ, 2 )= 1 p e (x µ2 )

29 Example The following normal distribution has mean μ = 100 and a standard deviation σ = 15

30 Standard normal distribution It is the normal distribution with a mean equal to 0 and a standard deviation (also variance) equal to 1: z N(0, 1) The standard normal distribution is often abbreviated to z. It is frequently used to simplify working with normal distributions.

31 Standard normal distribution

32 Reading a normal distribution

33 Sampling distribution of a statistic It is the distribution obtained by calculating the statistic (e.g. the mean) from an infinite number of independent samples of size n

34 Example An experiment measures the time it takes n = 10 people to visually locate a target on a computer screen. The same experiment is repeated a large (or infinite) number of times, where each time, we draw a new sample of size n. For each experiment, we compute the mean time: Experiment 1: M = 11.4 s Experiment 2: M = 12.6 s Experiment 3: M = 10.2 s... What s the distribution of these mean values?

35 Sampling distribution of a statistic Such distributions are interesting as they determine the probability of observing a particular value of the statistic, e.g., the mean. It is often very different than the distribution of the data used to calculate the statistic. distribution of the data sampling distribution of their means

36 Sampling distribution of the mean Its mean value is also the mean (expected value) of the original population the samples were drawn from Its standard deviation (SD) is known as the standard error of the mean (SEM)

37 The central limit theorem (CLT) States that the sampling distribution of a statistic approaches the normal distribution as n approaches infinity. It applies to statistics computed by summing or averaging quantities (means, variances) but not to standard deviations (squared root of an average)

38 The central limit theorem (CLT) States that the sampling distribution of a statistic approaches the normal distribution as n approaches infinity. It applies to statistics computed by summing or averaging quantities (means, variances) but not to standard deviations (squared root of an average) central = fundamental to probabilities and statistics limit = refers to a limit condition n!1

39 Practical importance of the CLT If the size of the sample is sufficiently large, then the sampling distribution of the statistic will be approximately normal (no matter what the distribution of the original population was) But which sample size is sufficiently large?

40 Sampling from normal distributions If the original population is normal, then the CLT will always hold, even if the sample size is as low as n = 1 The further the original population moves away from a normal distribution, the larger the sample size n should be

41 Sampling from binomial distributions Statistic of interest: Count of successes from n Bernoulli trials Frequency Frequency Frequency n = 10, P = n = 30, P = n = 100, P =.15 Frequency Frequency Frequency n = 10, P = n = 30, P = n = 100, P =.35

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43 Sampling from exponential distributions dexp(x) Distribution of source population Statistic of interest: Mean of a sample of n drawn from an exponential distribution Sampling distributions of the mean Frequency Frequency Frequency n= n= n=100

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45 What n is sufficiently large? Several textbooks claim that n = 30 is enough to assume that a sampling distribution is normal, irrespective of the shape of the source distribution. «This is untrue» [Baguley] There is no magic number to guarantee that.

46 Log-normal distribution A random variable X is log-normally distributed if the logarithm of X is normally distributed: X LogN(µ, 2 ) () ln(x) N(µ, 2 )

47 Simple math with logarithms log b (x) =a () b a = x log b (1) = 0 () b 0 =1 log b (b) =1() b 1 = b If the base of the logarithm is equal to the Euler number e = , we write: ln(x) =log e (x) Which we base to use is not important, but it is common to use e as a base. 47

48 Log-normal distribution A common choice for real-world data bounded by zero e.g., response time or task-completion time «The reasons governing frequency distributions in nature usually favor the log-normal, whereas people are in favor of the normal» «For small coefficients of variation, normal and log-normal distribution both fit well.» [ Limpert et al ]

49 Sampling from lognormal distributions dlnorm(x, meanlog = 0, sdlog = 1) Distribution of source population µ =0, = Sampling distributions of the mean Frequency Frequency Frequency n= n= n=100

50 Sampling from lognormal distributions Before... Sampling distributions of the mean Frequency Frequency Frequency n=10...and after applying a log transformation on the data n=30 n=100 Frequency Frequency Frequency n= n= n=100

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52 Skewed distributions Asymmetrical distributions are said to be skewed

53 The chi-square (χ 2 ) distribution Consider a squared observation z 2 drawn at random from the standard normal (z) distribution The distribution of z 2 will follow a χ 2 distribution with 1 degree of freedom (df)

54 The chi-square (χ 2 ) distribution Consider a squared observation z 2 drawn at random from the standard normal (z) distribution The distribution of z 2 will follow a χ 2 distribution with 1 degree of freedom (df) Probability density Probability density Z Z 2

55 The chi-square (χ 2 ) distribution A χ 2 distribution with k degrees of freedom is the distribution of a sum of squares of k independent variables that follow a standard normal distribution Q = kx Zi 2 =) Q i=1 2 (k)

56 The chi-square (χ 2 ) distribution Given the link between variances and sums of squares, the chi-square distribution is useful for modeling variances of samples from normal (or approximately normal) distributions.

57 The t distribution (Student s t) The sampling distribution of means for a normally distributed population Useful when: the sample size is small, and the population standard deviation is unknown published by William Gosset (1908) under the pseudonym «Student»

58 The t distribution (Student s t) When the population standard deviation σ is unknown and is estimated from the unbiased variance estimate: ˆ2 = np (x i ˆµ) i=1 n 1 then, the resulting standardized sample mean has a t distribution with ν = n - 1 degrees of freedom. Note: We ll further explain this later.

59 The t distribution (Student s t) A random variable X following a t distribution is denoted as: X t( ) Probability density t z Probability density t z t(1) t(29)

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61 R distribution functions Binomial distribution dbinom(x, n, P) Provides the probability mass function for the binomial distribution B(n,P) Examples: dbinom(4, 20,.2): It will return the probability of x = 4 successes for n = 20 Bernoulli trials with a P=.2 probability of success. dbinom(c(1,2,3,4), 10,.2): It will return a vector with the probabilities of x = {1, 2, 3, 4} successes for n =10 Bernoulli trials with a P =.2 probability of success.

62 R distribution functions Binomial distribution pbinom(x, n, P) Provides the cumulative probability mass function for the binomial distribution B(n,P) Example: pbinom(4, 20,.2): It will return the cumulative probability up to x = 4 successes for n = 20 Bernoulli trials with a P =.2 probability of success.

63 R distribution functions Binomial distribution rbinom(size, n, P) It will generate a random sample of size size from the binomial distribution B(n,P) Example: rbinom(10, 20,.2): It will return a random sample of size = 10 from the binomial distribution B(n = 20, P =.2)

64 R distribution functions Normal distribution dnorm(x, mean, sd) Provides the probability density function for the normal distribution with a mean value equal to mean and a standard deviation equal to sd. Examples: dnorm(.2, 0, 1): It will return the relative likelihood of the value x =.2, for the standard normal distribution. curve(dnorm(x, 100, 15), xlim = c(60, 140)): It will plot the probability density function from x = 60 to x = 140 for the normal distribution with mean = 100 and sd = 15.

65 R distribution functions Normal distribution pnorm(x, mean, sd) Provides the cumulative probability density function for the normal distribution with a mean value equal to mean and a standard deviation equal to sd. Example: pnorm(100, 100, 15): It will return the cumulative probability up to x = 100 for the normal distribution with mean = 100 and sd = 15. (What do you expect it to be?)

66 R distribution functions Normal distribution rnorm(size, mean, sd) It will generate a random sample of size size from the normal distribution with a mean value equal to mean and a standard deviation equal to sd. Example: rnorm(10, 0, 1): It will return a random sample of size = 10 from the standard normal distribution.

67 R distribution functions Binomial Normal Log-normal chi-squared Student Distribution function (pmf or cdf) Cumulative distr. function Random sampling dbinom(x, n, P) pbinom(x, n, P) rbinom(size, n, P) dnorm(x, mean, sd) dlnorm(x, mean, sd) pnorm(x, mean, sd) rnorm(size, mean, sd) plnorm(x, mean, sd) rlnorm(size, mean, sd) dchisq(x, k) pchisq(x, k) rchisq(size, k) dt(x, ν) pt(x, ν) rt(size, ν)

68 Intro to Confidence Intervals

69 Statistical inference The process of deducing the parameters of an underlying probability distribution from a sample Four broad types: point estimation interval estimation hypothesis testing prediction

70 Point estimates How much informative is the following graph? Mean Time (s) T1 T2 T3 Techniques

71 Point estimates A point estimate can be thought of as a «best guess» of the true population parameter Descriptive statistics such a the sample mean or the median are examples of point estimates Question: What are the point estimates of a population s variance and standard deviation?

72 Point estimates How much informative is the following graph? Mean Time (s) T1 T2 T3 Techniques A point estimate communicates no information about the uncertainty or quality of the estimate it provides

73 Interval estimate An interval estimate does not provide an exact value, but rather a range of values that the parameter might plausibly take. Most common method: constructing a confidence interval (CI)

74 Confidence interval (CI) It specify a range of values that is expected to contain the true parameter value (but it may not) true value It is associated with a confidence level, usually expressed as a percentage e.g., 95% CI or 99% CI

75 Formal interpretation of CIs Classical frequentists statistics view a probability as a statement about the frequency with which events occur in the long run. Of the many 95% CIs that might be constructed, 95% are expected to contain the true population parameter. The other 5% may completely fail!

76 Informal interpretation of CIs Formally speaking, a CI does not specify a probability range! A 95% CI does not necessarily contain the true population parameter with a 95% probability (or 95% confidence). However, it is often reasonable to treat a CI as an expression of confidence or belief that it does contain the true value. See [Baguley] and [Cumming and Finch, 2005] Attention: This view is not shared by everyone! It has received a lot of criticism by Bayesian statisticians.

77 Confidence level A 100% CI will include the whole range of possible values A 0% CI reduces to a point estimate A 95% CI is the most common choice (by tradition)

78 alpha If C is the confidence level of a CI, then: C = 100 (1 - α) where α (or alpha) represents the number of times that a C% CI is expected to fail: If C = 95, the α =.05

79 Structure of a confidence interval It is defined by two points that form its limits, i.e., its lower and upper bounds It can be symmetrical, where the point estimate lies in the center of the CI...or asymmetrical, where the point estimate is not at the center of the CI

80 Symmetrical CIs The intervals can be described by the point estimate plus or minus half of the interval, e.g., 165 ± 6 cm This half width of the interval is known as the margin of error (MOE)

81 Width of a CI Depends on the confidence level e.g., 99% CIs are wider than 95% CIs Also depends on the sampling distribution of the statistic The smaller the sample size, the wider the sampling distribution Small samples produce wide CIs

82 Example Consider the sampling distribution of the mean for a normally distributed population (M = 100, SD = 10) Frequency Frequency Frequency n= n= n=100 The sampling distribution becomes narrower as more samples are added.

83 In the following lecture, we will revisit confidence intervals and explain one can construct them. 83