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TNoms
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A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \sim N(\mu,\sigma^2)$$X_1,\dots,X_n \stackrel{iid}{\sim} N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic mean of the sample, how do I find the covariance between $$Cov(X_1,\overline{X})$$

I know (at least I think I do) that it should become equal to Var$(\overline{X})$, which I can state as $\frac{\sigma^2}{n}$ because $\sigma^2$ is known.

I rewrite as:

$$Cov(X_1,\overline{X})=E(X_1\overline{X})-E(X_1)E(\overline{X})$$

which becomes $E(X_1\overline{X})-E(\overline{X})^2$ but evaluating that first part is causing me some issues. Am I on the right track?

Thank you.

A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \sim N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic mean of the sample, how do I find the covariance between $$Cov(X_1,\overline{X})$$

I know (at least I think I do) that it should become equal to Var$(\overline{X})$, which I can state as $\frac{\sigma^2}{n}$ because $\sigma^2$ is known.

I rewrite as:

$$Cov(X_1,\overline{X})=E(X_1\overline{X})-E(X_1)E(\overline{X})$$

which becomes $E(X_1\overline{X})-E(\overline{X})^2$ but evaluating that first part is causing me some issues. Am I on the right track?

Thank you.

A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \stackrel{iid}{\sim} N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic mean of the sample, how do I find the covariance between $$Cov(X_1,\overline{X})$$

I know (at least I think I do) that it should become equal to Var$(\overline{X})$, which I can state as $\frac{\sigma^2}{n}$ because $\sigma^2$ is known.

I rewrite as:

$$Cov(X_1,\overline{X})=E(X_1\overline{X})-E(X_1)E(\overline{X})$$

which becomes $E(X_1\overline{X})-E(\overline{X})^2$ but evaluating that first part is causing me some issues. Am I on the right track?

Thank you.

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TNoms
TNoms
  • 95
  • 5

A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \sim N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic mean of the sample, how do I find the covariance between $$Cov(X_1,\overline{X})$$

I know (at least I think I do) that it should become equal to Var$(\overline{X})$, which I can state as $\frac{\sigma^2}{n}$ because $\sigma^2$ is known.

I rewrite as:

$$Cov(X_1,\overline{X})=E(X\overline{X})-E(X)E(\overline{X})$$$$Cov(X_1,\overline{X})=E(X_1\overline{X})-E(X_1)E(\overline{X})$$

which becomes $E(X\overline{X})-\overline{X}^2$$E(X_1\overline{X})-E(\overline{X})^2$ but evaluating that first part is causing me some issues. Am I on the right track?

Thank you.

A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \sim N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic mean of the sample, how do I find the covariance between $$Cov(X_1,\overline{X})$$

I know (at least I think I do) that it should become equal to Var$(\overline{X})$, which I can state as $\frac{\sigma^2}{n}$ because $\sigma^2$ is known.

I rewrite as:

$$Cov(X_1,\overline{X})=E(X\overline{X})-E(X)E(\overline{X})$$

which becomes $E(X\overline{X})-\overline{X}^2$ but evaluating that first part is causing me some issues. Am I on the right track?

Thank you.

A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \sim N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic mean of the sample, how do I find the covariance between $$Cov(X_1,\overline{X})$$

I know (at least I think I do) that it should become equal to Var$(\overline{X})$, which I can state as $\frac{\sigma^2}{n}$ because $\sigma^2$ is known.

I rewrite as:

$$Cov(X_1,\overline{X})=E(X_1\overline{X})-E(X_1)E(\overline{X})$$

which becomes $E(X_1\overline{X})-E(\overline{X})^2$ but evaluating that first part is causing me some issues. Am I on the right track?

Thank you.

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TNoms
TNoms
  • 95
  • 5

Covariance between partitions of a normal distribution

A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \sim N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic mean of the sample, how do I find the covariance between $$Cov(X_1,\overline{X})$$

I know (at least I think I do) that it should become equal to Var$(\overline{X})$, which I can state as $\frac{\sigma^2}{n}$ because $\sigma^2$ is known.

I rewrite as:

$$Cov(X_1,\overline{X})=E(X\overline{X})-E(X)E(\overline{X})$$

which becomes $E(X\overline{X})-\overline{X}^2$ but evaluating that first part is causing me some issues. Am I on the right track?

Thank you.