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Chapter 5: Q24E (page 346)

Suppose that\({X_1}and\,{X_2}\) have the bivariate normal distribution with means\({\mu _1}and\,{\mu _2}\) variances\({\sigma _1}^2and\,{\sigma _2}^2\), and correlation 蚁. Determine the distribution of\({X_1} - 3{X_2}\).

Short Answer

Expert verified

The random variable \({X_1} - 3{X_2}\)has the normal distribution with mean\({\mu _1} - 3\,{\mu _2}\)and variance\({\sigma _1}^2 + 9{\sigma _2}^2 - 6\rho {\sigma _1}{\sigma _2}\)

Step by step solution

01

Step 1:Given information

\({X_1}{\rm{and}}\,{X_2}\) have the bivariate normal distribution with means\({\mu _1}{\rm{and}}\,{\mu _2}\) variances\({\sigma _1}^2{\rm{and}}\,{\sigma _2}^2\), and correlation 蚁. We need to find out the distribution of \({X_1} - 3{X_2}\).

02

Step-2: Distribution of\({X_1} - 3{X_2}\).

Linear combination of bivariate normal formula: Two random variables \({X_1}{\rm{and}}\,{X_2}\)have a bivariate normal distribution, Let Y =\({a_1}{X_1} + {a_2}{X_2} + b\), where \({a_1},{a_2}\) and b are arbitrary given constants. Then Y has the normal distribution with mean \({a_1}{\mu _1} + {a_2}{\mu _2} + b\) and variance\({a_1}^2{\sigma _1}^2 + {a_2}^2{\sigma _2}^2 + 2{a_1}{a_2}\rho {\sigma _1}{\sigma _2}\)

\({a_1} = 1,{a_2} = - 3,b = 0\), putting these value with the help of the above formula we get that the random variable \({X_1} - 3{X_2}\)has the normal distribution with mean\({\mu _1} - 3\,{\mu _2}\)and variance \({\sigma _1}^2 + 9{\sigma _2}^2 - 6\rho {\sigma _1}{\sigma _2}\)

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Most popular questions from this chapter

Suppose that two players A and B are trying to throw a basketball through a hoop. The probability that player A will succeed on any given throw is p, and he throws until he has succeeded r times. The probability that player B will succeed on any given throw is mp, where m is a given integer (m = 2, 3, . . .) such that mp < 1, and she throws until she has succeeded mr times.

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