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Random Variables Numeric outcomes of a random process. Probability Distribution, mean, standard deviation... Theoretical Results The value of a random variable will vary from trial to trial. Each trial or sampled value is a random variable. We take sample results and combine to find totals, means, variances, and standard deviations. Section 3.4 Notation So far we can define random variables: discrete and continuous Y X probability mean variance standard deviation We sample from popluations. Suppose a machine produces metal rods. The random variable X is the length. If we can predict where the sample means cluster we have a way to gauge how plausible our assumption was for the population. Consider Random Variable X: Add a constant to create a new random variable. What happens to the mean and standard deviation? Let X represents a randomly selected annual salary at a particular company. Suppose each employee at the company gets a $5000 endofyear bonus. This is a new random variable! Consider Random Variable X: Multiply by a constant to create a new random variable. What happens to the mean and standard deviation? Let X represents the diameter of a randomly selected bolt. If we are interested in the circumference then we can create a new random variable and predict its mean, variance, and standard deviation. Linear Combinations of INDEPENDENT Random Variables: How is standard deviation affected? X1, X2, ..., Xn X1+X2+ ...+Xn X1, X2, ..., Xn cX1+cX2+ ...+cXn X and Y X Y Wood beams are made up of three layers. Layers are a simple random sample from a population of boards. Beam = x + x + x The beam thicknesses are a new random variable that clusters around its own mean and has its own variation. Let's predict what those are... all pieces came from the same original population so they each have the same mean Notice that the standard deviation of the new population of constructed beams grew...uncertainty in total height grew compared to the uncertainty we assumed from the population of pieces. Example: Three steel rods are going to be assembled end to end. Length of the first component: Length of the second component: Length of the third component: 5.1cm 0.2cm 4.3 cm 0.1cm 0.5 cm 0.09cm Once assembled, find an estimate of the mean total length and the uncertainty (standard deviation) of the total length. All possible assembled components will have a total length that clusters around 9.9 cm. Notice that the uncertainty (standard deviation) of the assembled parts has grown compared to the individual three populations of components. Once again variation grows!! Often in statistics we take a sample (SRSindependent) and we find the mean of the values in the sample. Population Take a sample. All sampled values come from the same population. Typically we take one sample from a population. In theory if we take many samples and find all the sample means, then we would see them cluster around a mean (this is the mean of all possible sample means) and they would have a spread (this is the standard deviation of all the sample means). We can determine these. The sample mean is a new random variable. So to sum up...this is a big deal!! How can we reduce standard deviation (uncertainty)...? Take several INDEPENDENT measurements and find the mean. We create a new random variable that has its own mean and standard deviation. The sample means cluster around the population mean which makes the sample mean our best guess as to the value of the population mean. The mean of several measurements has the same accuracy as a single measurement BUT is more precise. Example: I know a scale produces measurements with uncertainty 0.2g. We want to use the scale to measure the mass of an object. (1) If we make ONE measurement, then the uncertainty does not change. (2) If we make five measurements and find the sample mean, then we can decrease the uncertainty.
