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5-1 Random Variables and Probability Distributions The Binomial Distribution Random Variables Discrete – These variables take on a finite number of values, or a countable number of values Number of days absent Number of students taking a course Continuous – These variables can take on an infinite number of values on a number line Time it takes for students to drive home Gallons of gas you buy each time you go to the gas station (Typically length, temperature, volume, time, etc) Probability Distribution A probability distribution is an assignment of probabilities to the specific values of a random variable, or to a range of values of the random variable. Discrete: probability assigned to each value of the random variable (and the sum = 1) What does this mean? Lets look at a graph of a probability distribution of the discrete model, and see what it is: 30.00 25.00 20.00 Series1 15.00 10.00 5.00 0.00 1 2 3 4 5 6 7 8 9 10 How do we look at it? 30.00 25.00 20.00 Series1 15.00 10.00 5.00 0.00 1 2 3 4 5 6 7 8 9 10 It looks like A histogram, where the height is the relative probability (i.e. a RELATIVE FREQUENCY) and the bin is the particular number. Does this look familiar?? So what is the probability of choosing a 7 or a 3? P(7 or 3) = Mean and Standard Deviation Probability distributions have a mean and standard deviation. For discrete population probability distributions, the mean and the standard deviation are given by formulas… Which letters will we use? and s or μ and σ? Population…. Anyone?? Mean and Standard Deviation μ xP(x) σ (x μ) P(x) 2 Sometimes this is called the expected value of a distribution - it is an AVERAGE value, or what can be thought of as a central point (cluster point) Where x is the value of the random variable, P(x) is the probability of that variable and The sum is taken for all the values of that random variable. Notice – we are now discussing mean and standard deviation of something other than = Risk – the likelihood that a random variable is raw data… different from the mean Lets consider the ways you can toss a coin four times Assume the coin is balanced (i.e. H and T equally likely) Assume also there is no memory (the coin doesn’t remember that the last toss was heads). There are sixteen outcomes, right? (1/2)4 Lets draw out ALL possible outcomes To assign a discrete random variable Let x = heads, so x=0 TTTT x=1 HTTT, THTT, TTHT, TTTH x=2 HHTT, HTHT, HTTH, THHT, THTH, TTHH x=3 HHHT, HHTH, HTHH, THHH x=4 HHHH So the probability model would be… # of heads 0 1 2 3 4 Probability .0625 .25 .375 .25 .0625 Calculate the mean and SD of the distribution mean = 2 Standard Deviation = 1 Linear Functions of Random Variables Suppose I have a and b, which are constants. A new function L = a + bx (where x is a random variable) ALSO has a mean, variance and standard deviation. Linear Functions of Random Variables Suppose I have a and b, which are constants. A new function L = a + bx (where x is a random variable) ALSO has a mean, variance and standard deviation. μL a bμ 2 L 2 σ b σ σL b σ 2 Combining independent random variables To make a linear combination of two independent random variables x1 and x2 , W = ax1 + bx2 and Combining independent random variables To make a linear combination of two independent random variables x1 and x2 , W = ax1 + bx2 and μW aμ1 bμ2 2 W 2 2 1 2 σ a σ b σ 2 2 1 2 2 2 σW a σ b σ 2 2 What? All this is a way to look what happens when you transform data (for instance, if I take all the data and multiply by 2 then add 10, for rescaling purposes). Application The manager of a computer company quickly shipped 2 computers to a client on the same day as the order. Unfortunately, the two computers were accidentally chosen from a stockroom with an inventory of 15 computers, 4 of which were refurbished. If one of the computer is refurbished it will be sent back at your expense ($100). If both are refurbished, the client will cancel the order this month and you will lose $1000. What is the expected value and standard deviation of your loss? Sources http://www.mathworks.com/access/helpdesk/help/toolbox/stats /index.html?/access/helpdesk/help/toolbox/stats/f4218.htm