Linear combinations of random variables
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Linear combinations of random variables
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In pure mathematics we are often concerned with the effect of a transformation on a function &- how a translation or a scaling affects the graph of the function. We are also concerned with the addition of functions to give new functions. Translations and scaling occur naturally within the context of random probability distributions. Likewise, we are also interested in the question of what is the probability distribution that arises when two probability distributions are added together or one is subtracted from the other. For example, suppose a company uses a machine to produce rods with average diameter d. They know the probability distribution of actual diameter, D, and the want to know what will be the average diameter and probability distribution of the diameters if they reset the machine to double the diameters. Another example concerns the addition of examination scores. An exam consists of two parts, and scores in each will be given by separate probability distributions. What is the probability of the combined scores?
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Contents of
Linear combinations of random variables
1 Linear combinations of random variables
2 Independence
3 Scaling and translation of random variable X
4 Proof, for any discrete random probability distribution in general
5 Proof for any continuous random probability distribution
6 Expectation and variance of the linear combination of random variables
7 Form of the linear combination of random variables
8 Scaling and translation of a Normal distribution
9 Linear Combinations of independent Normal distributions
10 Linear combination of independent Poisson distributions.
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