Given that the random variables $X$ and $Y$ have distributions $N(60, 25)$ and $N(54, 64)$ r

Question

Given that the random variables $X$ and $Y$ have distributions $N(60, 25)$ and $N(54, 64)$ respectively, and that $\bar{X}$ and $\bar{Y}$ denote the means of $7$ independent observations of $X$ and $11$ independent observations of $Y$ respectively. State the sampling distributions of $\bar{X}-\bar{Y}$ $2\bar{X}+5\bar{Y}$ $4\bar{X}-3\bar{Y}$ $\frac{{{X}_{1}}+{{X}_{2}}+{{X}_{3}}-{{Y}_{1}}-{{Y}_{2}}-...-{{Y}_{5}}}{8}$ $\frac{{{X}_{1}}+{{X}_{2}}+\ldots +{{X}_{10}}+{{Y}_{1}}+{{Y}_{2}}+...+{{Y}_{5}}}{15}$

Accepted Answer

(i) $N\left(6,\frac{723}{77}\right)$ (ii) $N\left(390,\frac{12300}{77}\right)$ (iii) $N\left(78,\frac{8432}{77}\right)$ (iv) $N\left(-\frac{45}{4},\frac{395}{64}\right)$ (v) $N\left(58,\frac{38}{15}\right)$

Linear Combinations of Independent Normal Sample Means

Linear Combinations of Independent Normal Sample Means

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