Linear combinations of random variables
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Scaling and translation of random variable X
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Let X be a random variable with expectation E(X) and variance Var(X). Let a and b be constants. Then [Equation goes here &- download the original to see it.] The expected value of X &- that is, the expected central tendency &- is translated by the addition of b and multiplied by the scale factor a. The variance is not affected by a translation, but it is more dispersed by the scaling, if a > 1, or less dispersed if a < 1. [Diagram goes here &- download the original to see it.] We will prove this result for three separate cases. Proof, for a discrete random probability distribution where the probability of each observation is equal. In order to prove these results we recall that [Equation goes here &- download the original to see it.] Then [Equation goes here &- download the original to see it.] Note here that Sb means &'add be n times&', hence Sb = nb. Also, [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] Independence is not an issue here since we are only dealing with one variable. Before proceeding to the other proofs we illustrate the use of these formulae. Example The random variable X has mean 20 and variance 6. What are the mean and variance of the random variables [Equation goes here &- download the original to see it.] Answer [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.]
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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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