University of Central Florida School of Electrical Engineering and Computer Science EGN 3420 Engineering Analysis Fall 2009 dcm Probability and Statistics Concepts Random Variable a rule that assigns a numerical value to each possible outcome of an experiment All possible outcomes of the experiment constitute a sample space A random variable X on a sample space S is a function X S 7 R which assigns a real number X s to every sample point s S This real number is called the probability of that outcome A discrete random variable maps events to values of a countable set e g the set of integers each value in the range has a probability greater than or equal to zero Example 1 the experiment is a coin toss the outcome is either 0 head or 1 tail If the coin is fair then p0 p1 0 5 this means that in a large number of coin tosses we are likely to observe heads in about half of the cases and tails in the other half of the cases Another example when you throw throw a dice the outcome could be 1 2 3 4 5 or 6 for a fair dice p1 p2 p3 p4 p5 p6 1 6 A continuous random variable maps events to values of an uncountable set e g the real numbers Example 2 the experiment is to measure the speed of cars passing through an intersection the speed could be any value between 15 and 80 miles hour the probability of observing cars with a speed of 19 1 miles hour could be zero but the probability of observing cars with a speed from 15 to 19 1 miles hour could be P19 1 0 3 which means that 30 of the cars we observed have a speed in the range we considered A discrete random variable X has an associated probability density function also called probability mass function pX x defined as pX x Prob X x and a probability distribution function also called cumulative distribution function PX x defined as X PX t Prob X t pX x x t Example 3 You have a binary random variable X the outcome is either 0 or 1 and p0 Prob X 0 q and p1 Prob X 1 p with p q 1 Bernouli trials call the outcome of 1 a success and ask the question what is the probability Yn that in n Bernoulli trials we have k successes n k n pk Prob Yn k p 1 p n k pk 1 p n k k k n k The binomial cumulative distribution function is B t n p t X n k 0 k pk 1 p n k Example 4 You have again Bernoulli trials and ask the question how many trails you need before the first success If the first success occurs at the i th trial then pZ i q i 1 p This is called a geometric distribution It is easy to prove that X i 0 q i 1 p p 1 1 q A continuous random variable X has an associated probability density function also called probability mass function pX x defined as fX x Prob X x and a probability distribution function also called cumulative distribution function FX x defined as Z t FX t Prob X t fX x dx The expectation of random variable X E X is defined by P i xi pX xi if X is discrete E X R xf x dx if X is continuous The variance Var X and standard deviation of random variable X are defined by P 2 i xi E X pX xi if X is discrete Var X 2 R x E X 2 f x dx if X is continuous The moment of order k of random variable X is defined as P k if X is discrete i xi pX xi k E X R k x f x dx if X is continuous The centered moment of order k of random variable X is defined as the k th moment of the random variable x E X P k i xi E X pX xi if X is discrete k E X E X k R x E X k f x dx if X is continuous Examples of common distributions 1 Uniform distribution in the interval a b see Figure 1 1 if a x b b a fX x 0 if x a or x b if x a 0 x a if a x b FX x b a 1 if x b 1 a b a b 1 Figure 1 Probability density function PDF and cumulative distribution function of a uniform distribution 2 Standard normal distribution 1 2 x e x 2 3 Normal distribution with mean and standard deviation f x 1 2 2 e x 2 2 The probability density function PDF and the cumulative distribution function CDF of a normal distribution are displayed in Figures 2 and 3 1 0 0 0 0 2 0 8 2 0 2 2 1 0 2 5 0 2 0 5 0 6 0 4 0 2 0 0 5 4 3 2 1 0 1 2 3 4 5 x Figure 2 Probability density function PDF of a normal distribution 1 0 0 8 0 0 0 2 2 0 2 2 1 0 2 5 0 2 0 5 0 6 0 4 0 2 0 0 5 4 3 2 1 0 1 2 3 4 5 x Figure 3 Cumulative distribution function CDF of a normal distribution 4 Exponential distribution with parameter The probability density function PDF f x and the cumulative distribution function CDF F x of an exponential distribution are displayed in Figures 4 and 5 f x F x xe x if x 0 0 if X 0 1 e x if x 0 0 if X 0 Figure 4 Probability density function PDF of an exponential distribution Figure 5 Cumulative distribution function CDF of an exponential distribution The use of Matlab for generating different http www aquaphoenix com lecture matlab10 page2 html distributions is from