Understanding Random Variables: Probability, Moments, and Distributions - Prof. 1435, Apuntes de Psicología

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““TTécnicas de investigaciónécnicas de investigación””

Academic course 2012-2013 Academic course 2012-

• Solanas, A., Salafranca, Ll., Fauquet, J. y Núñez, M. I. (2005). Estadística descriptiva en ciencias del comportamiento. Madrid: Thomson. Where: 4/2 EST What: Capítulo 3

*No^ perfect^ prediction^ of^ human^ behavior (empirical reality): uncertainty even when the same stimuli. *The^ surprising^ component^ of^ reality^ can^ be called “random” or “stochastic”. It is part of our lives and also of science. *Randomness is expressed when throwing a dice and when thinking about a person’s behavior. *Random variables are an attempt to describe and formalize mathematically what is random: quantify uncertainty & get information from what is unpredictable.

*A random variable is a correspondence between: a) events that can occur when a random experiences is taking place, e.g., the results when tossing a coin. b) real numbers, which are infinite. *Example:^ a^ coin^ is^ tossed^10 times^ and^ the frequency of heads is tallied. Suppose the results is

7. A correspondence has been made between the result of a random experience and a real number obtained counting the amount of heads. *A^ random^ variable^ requires^ knowing^ the probabilities of each of the possible results!!! (e.g., a student answering at random). Or perform an empirical study (prob of a number of correct answers not at rrandom?)

*There is a single sequence of results leading to 0 heads, but there are different sequences leading to 5 heads. Thus, it is not as common to obtain 0 heads as it is to obtain 5 heads in 10 tosses. *So the random variable “number of heads” requires a quantification of the degree of certainty we have for each of the possible results. *Probability is the concept used for quantifying the differential frequency of each result of the random experience. Result of the random experience  corresponding real number  probability assigned.

For a single random variable:

• Discrete (a bijective function; one-to-one correspondence to the

natural numbers)

o (^) Finite: a finite amount of numbers is possible o (^) Infinite : a finite amount of numbers is not possible

• Continuous^ (a^ bijective^ function^ does^ not^ exist:^ in^ any^ interval

infinite amount of values)

o (^) Infinite totally continuous (there is no discontinuity in the

distribution function)

o Infinite partially continuous (there is at least one discontinuity in

the distribution function)

According to the amount of random variables:

• One-dimensional (a single random variable)
• Two-dimensional (a joint distribution of two random variables)
• n-dimensional

Event prob.,Trials 0,2,

Binomial Distribution

x

probability 0 5 10 15 20 25

0,2 Event prob. 0,

Geometric Distribution

x

probability 0 2 4 6 8 10

Event prob.,Successes 0,45,

Negative Binomial Distribution

x

probability 0 10 20 30 40 50

0,08 Mean 3

Poisson Distribution

x

probability 0 2 4 6 8 10 12

What’s in

between?

*Density: informs about the amount of values in an interval. Density ≠ probability. *The probability is obtained for an interval of values centered at x *The^ probability^ of^ an^ individual^ value^ is assumed to be zero. / 2 / 2 ( ) ( ) x dx x dx p x f x dx    

*Discrete^ variable:^ It^ refers^ to^ cumulative probability: sum of mass probabilities of individual values. *Continuous^ variable:^ It^ refers^ to^ cumulative probability: sum of densities or sum of probabilities of intervals of values. F k ( ) Prob( X  k ) ( ) ( ) x F x f x dx    

Event prob.,Trials 0,1,

Binomial Distribution

x

cumulative probability 0 2 4 6 8 10

1 Event prob. 0,

Geometric Distribution

x

cumulative probability 0 20 40 60 80

Event prob.,Successes 0,1,

Negative Binomial Distribution

x

cumulative probability 0 50 100 150 200 250

1 Lower limit,Upper limit 0,

Discrete Uniform Distribution

x

cumulative probability 0 0,4 0,8 1,2 1,6 2

A line?

What’s in

between?

*The moments of a random variable are global indicators of some of its characteristics, unlike mass probability, density and distribution function which give information about each of the values/intervals. *Other global indicators: based on position (quantiles). Mathematical expectancy: information about location. Variance: information about scatter. Skewness: information about shape; equally distant from the mean lower and higher values? Kurtosis: information about shape; as peaky as the Normal distribution? More? Less?

*A^ k -order non-centered moment is defined for discrete

random variables as

*A^ k -order moment centered with respect to the arbitrary

point c is defined for discrete random variables as

*In both expressions^ n^ designates the amount of values

of the random variable.

  ^ 

1

Prob

n k k k (^) i i i

m E^ X X^ x^ x



 ^   ^ ^ ^ 

' 1

Prob

n k k k i i i

 E X c X x x c



*It is the expected (≠ certain) value of the random variable in a trial.

• Informs about the location (balance point) of the random variable values. *Properties of mathematical expectancy If all values of a random variable are non- negative, then E(X)  0. E(X – E(X)) = 0. If a and b are constants, E(aX + b) = aE(X) + b: affected by changes in location and scale.

• 4 questions: true/false^ ^ Math expectancy (Correct=2) *0 correct (also all correct):

1 result

*1 correct (also all but one correct):

4 results

*2 correct: 6 results      