AEM 201 1st Edition Lecture 12PREVIOUS LECTUREI. Some Important Discrete Probability DistributionsCURRENT LECTUREI. Continuous Probability DistributionsII. Some Important Continuous Probability DistributionsCONTINUOUS PROBABILITY DISTRUBTIONS- Continuous Random Variable: a numerical description of the outcome of an experiment whose outcome can assume any numerical value in an interval or a collection of intervalsSOME IMPORTATNT CONTINUOUS PROBABILITY DISTRIBUTIONS- Continuous Uniform Probability: expresses the likelihood of outcomes for continuous random variable x for which all outcomes are equally likely- *- Where a=minimum possible value of random variable x- Where b=maximum possible value of random variable x- Characteristics of a continuous uniform probability distributiono The random variable can assume any value within a rangeo All possible values within this range are equally likely o No values outside of the range can occuro Mean expected value is: E(x)=(a+b)/2o The variance equals (b-a)^2/n- The area under ANY probability curve equals 1- This leads to two conclusions about working with probability distribution functions for continuous variableso P(x)=0 for any given value of xo We can only talk meaningfully about the probability of a range of of values- So how can we calculate probabilities for continuous random variables over ranges- To find P(a<=x<=b) we integrate the probability distribution function f(x)from a tobe i.e.- Exponential Probability Distribution: expresses the likelihood of outcomes for a continuous random variable x that represents the amount of time or space that passes between consecutive occurrences of a Poisson random variable. o *- Characteristics of exponential probability distributions are:o The random variable can assume any positive valueo The random variable represents the amount of the interval (time, space etc.) that passes between consecutive occurrences of a Poisson evento Mean is E(x)=o Variance=the mean- The slope is changing at a constant rate between 0 and infinity- B and must be in the same units- Normal Probability Distribution: expresses the likelihoods of outcomes for a continuous random variable x with a particular symmetric and unimodal distribution- The slope is changing at a changing rate between negative infinity and infinity- Characteristics of the normal probability distribution are:o There are an infinite number of normal distributions each defined by the unique combo of the mean and standard deviationo  determines the central location and variance determines the spread and widtho The spread is symmetric about the meano It is unimodalo It is asymptotic with respect to the horizontal axiso The area under the curve is 1o It is neither platykurtic nor leptokurtico It follows the empirical rule-rather defines it- Normal distributions with different variances have different heights- Normal distributions with different means have are shifted - If we pick the mean to be 0 and the standard deviation to be 1, they become irrelevant to the math- The area at a point =0 but the area of a range has