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lerem456

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Let $X_1, ..., X_{35}$ be independent Poisson random variables having mean and variance 2.

Let $Y_1, ..., Y_{15}$ be independent Normal random variables having mean 1 and variance 2.

(a.) Specify the (approximate) distributions of $\bar{X}$.

(b.) Find the probability $P(1.8 \leq \bar{X} \leq 2.3)$

(c.) Specify the (approximate) distributions of $\bar{Y}$.

(d.) Specify the (approximate distributions of $\bar{X} + \bar{Y}$.

(e.) Find the probability $P(Y_1 + ... + Y_{15} \leq 20)$.

My attempt,

(a.) Since $n > 30$. I can use the central limit theorem to state $\bar{X} \sim N(2, \frac{2}{35})$

(b.) Apparently this one is incorrect and I'm unsure where my error is.

\begin{equation*}\sigma = \sqrt{\frac{2}{35}} = 0.24\end{equation*}

\begin{equation*}P(-\frac{0.2}{0.24} \leq Z \leq \frac{0.3}{0.24}) = P(-0.84 \leq Z \leq 1.25)\end{equation*}

(c.) $\bar{Y} \sim N(1, \frac{2}{15})$

(d.) $\bar{X} + \bar{Y} ~N(35*2 + 15*1, 35*\frac{2}{35} + 15*\frac{2}{15}) = \sim N(50, 4)$

(e.) $P(Z \leq \frac{5}{\sqrt{30}}) = P(Z \leq 0.91)$Any insight on any part is greatly appreciated. Thanks.

Let $Y_1, ..., Y_{15}$ be independent Normal random variables having mean 1 and variance 2.

(a.) Specify the (approximate) distributions of $\bar{X}$.

(b.) Find the probability $P(1.8 \leq \bar{X} \leq 2.3)$

(c.) Specify the (approximate) distributions of $\bar{Y}$.

(d.) Specify the (approximate distributions of $\bar{X} + \bar{Y}$.

(e.) Find the probability $P(Y_1 + ... + Y_{15} \leq 20)$.

My attempt,

(a.) Since $n > 30$. I can use the central limit theorem to state $\bar{X} \sim N(2, \frac{2}{35})$

(b.) Apparently this one is incorrect and I'm unsure where my error is.

\begin{equation*}\sigma = \sqrt{\frac{2}{35}} = 0.24\end{equation*}

\begin{equation*}P(-\frac{0.2}{0.24} \leq Z \leq \frac{0.3}{0.24}) = P(-0.84 \leq Z \leq 1.25)\end{equation*}

(c.) $\bar{Y} \sim N(1, \frac{2}{15})$

(d.) $\bar{X} + \bar{Y} ~N(35*2 + 15*1, 35*\frac{2}{35} + 15*\frac{2}{15}) = \sim N(50, 4)$

(e.) $P(Z \leq \frac{5}{\sqrt{30}}) = P(Z \leq 0.91)$Any insight on any part is greatly appreciated. Thanks.

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