RANDOM VARIABLES 1 13 Recall that a random process is a series of independent trials trials can consist of events or selections where the outcome of each trial is unpredictable but in the long run a pattern of outcomes emerges that is a distribution emerges That distribution gives us a probability model that can be used in data analysis A random variable is a function that maps each trial of a random process to a quantitative value Put another way a random variable extracts a quantity of some sort from each trial in a random process The sample space of a random variable is the set of all possible values of the random variable Random Process Sample Space of the random variable X trial N trial trial Number Number number Examples of random variables Trial randomly select a student randomly select two whole numbers randomly select a class at FSU randomly select three students randomly select four persons Numerical value extracted their height or their weight the maximum of the two the number of students in the class the number of them who got a flu shot last year the proportion of them who were born in the morning randomly select five cars on the highway their average speed randomly select two books randomly select a bus route randomly select a person eating a burger the difference between the numbers of pages they have the total distance the time they take to finish the burger 2014 Radha Bose Florida State University Department of Statistics D C C D D D D C D C C RANDOM VARIABLES 2 13 C the time it takes for the engine to warm up after it is started randomly select a stationary car in cold weather A discrete random variable is one where the sample space consists of discrete values This means that the values are distinct and countable that is you can either write them all down finite number of values or you can begin the list and show a pattern even if you cannot finish the list infinite number of values The probability distribution of a discrete random variable is given by a probability mass function pmf which is graphed as a probability histogram The total area under a probability histogram is always 1 Quick Illustration Sample Space of the random variable X Probability of each value 2 1 2 3 PMF 1 3 r S P X r 1 r 6 1 6 6 6 1 5 6 4 6 2 3 3 6 1 2 2 6 1 3 1 6 0 1 2 3 4 5 6 7 2014 Radha Bose Florida State University Department of Statistics RANDOM VARIABLES 3 13 Real life Examples number of objects age to the nearest year rounded measurements of height weight temperature duration distance length area volume etc A continuous random variable is one where the sample space consists of values that are measured on a continuous scale What this means is that no matter which two possible values you choose there will always be a third possible value between them Thus there will be an infinite number of possible values so you cannot write them all down and you cannot even begin a list and show a pattern You can only specify a lower bound for example all real numbers greater than 50 or an upper bound for example all real numbers less than 60 or both for example all real numbers between 30 and 40 or neither for example all real numbers The probability distribution of a continuous random variable is given by a probability density function pdf which is graphed as a density curve A density curve never goes below the horizontal axis the data axis and the total area under the curve is 1 Any curve that has these two properties qualifies as a density curve and the equation of the curve qualifies as a PDF However PDFs do not give us probabilities directly the way PMFs do Instead area under the density curve gives us probabilities the PDF needs to be integrated over an interval in order to get the probability of that interval integrating a function over an interval gives us the area under the curve above that interval Integrating a function over a single value results in an area of zero therefore the probability of an individual value is considered to be zero Continuous random variables are sometimes used to approximate discrete random variables since the former are often easier to manipulate Real life Examples unrounded measurements of height weight temperature duration distance length area volume etc Rounded measurements would be discrete 2014 Radha Bose Florida State University Department of Statistics RANDOM VARIABLES 4 13 When min and max exist F min P min F max 1 The cumulative distribution function cdf of a random variable X is a function F which when evaluated at a specific value r gives the probability that the random variable is less than or equal to r F r P X r For any value r in the sample space Discrete F r the pmf summed over the set x x r Continuous F r the pdf integrated over the interval r For any value r in the sample space P X r 1 F r For any two values c and d in the sample space where c d P c X d F d F c The expected value of a random variable is a weighted average of the values where the weighting comes from the probability distribution The expected value is the mean of the random variable The standard deviation of a random variable can similarly be considered to be a weighted average of the distances from the mean For a finite discrete random variable X that has n values x1 x2 xn values of the random variable p1 p2 pn the probabilities of occurrence of x1 x2 xn respectively Expected Value E X xp x1p1 x2p2 xnpn Mean X same as the expected value Standard Deviation X sqrt x 2p sqrt x1 2p1 x2 2p2 xn 2pn For a continuous random variable X with pdf f x Expected value is E X xf x dx and Variance is Var X x 2f x dx where both integrals are evaluated from to The standard deviation would just be the square root of the variance 2014 Radha Bose Florida State University Department of Statistics RANDOM VARIABLES 5 13 10 6 6 7 7 7 8 8 8 8 9 n 10 6 6 7 7 7 8 8 8 8 9 10 7 4 For some known families of random variables there are shortcut formulas for obtaining the expected value and standard deviation WEIGHTED AVERAGE List of values Average 6 2 7 3 8 4 9 1 7 4 6 2 7 3 8 4 9 1 7 4 10 6 2 7 3 8 4 9 1 7 4 n could be any natural number and the average would still be 7 4 provided that 20 of the list consisted of 6 s 30 of the list consisted of 7 s 40 of the list consisted of …