Linear combinations of random variables
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Linear combination of independent Poisson distributions.
If X and Y have independent Poisson distributions, then X+Y has a poisson distribution. [Equation goes here ; ; download the original to see it.] Example Drilling on a certain exploration oil rig can stop for two reasons. Firstly, the drill head can be broken. Secondly, the drilling strikes an underground lake. Stoppages due to broken drill heads have Poisson distribution with mean per month. Stoppages due to underground water have Poisson distribution with mean 1 per month. The two distributions are independent. Find (i) the distribution of the total stoppages per month; (ii) the probability that, in a given month, there are no more than two stoppages. (i)Let X represent the number of stoppages due to broken drill heads, and Y the number due to underground water. [Equation goes here ; ; download the original to see it.] [Equation goes here ; ; download the original to see it.] (ii) For any Poisson distribution [Equation goes here ; ; download the original to see it.] Here [Equation goes here ; ; download the original to see it.] Therefore [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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