Linear combinations of random variables
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Independence
In this section we are concerned with the translation, scaling and addition of independent random variables. What does the term "independent" mean? It means that observation of one variable, say X, does not affect the probability distribution of observation of another, say Y. Two events, A & B, are independent if the probability of both A and B occurring together is equal to the probability of A occurring multiplied by the probability of B occurring. [Equation goes here &- download the original to see it.] Let X and Y be random variables. Let X = xi and Y = yi be the events X takes the value xi; and Y takes the value yi respectively. Then X and Y are independent if [Equation goes here &- download the original to see it.] That is, the probability that X takes the value xi and Y takes the value yi is equal to the probability that X takes the value xi multiplied by the probability that Y takes the value yi. Some of the results stated in this section apply regardless of whether the variables are dependent of independent. However, we are working generally within the context of independent random variables of different distributions. There is a second distinction that should be kept in mind&: 1.The case where we wish to combine in someway the probability distributions of two separate independent random variables X and Y. 2.The case where we wish to combine in some way two independent observations, X1 amd X2, of the same random variable X. The two cases are not the same, and the theoretical results are different. Here we deal only with the first case.
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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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