I am trying to figure out the covariance between sample mean and sample STANDARD DEVIATION (NOT VARIANCE) from a population. We DO NOT know whether the population is normal (if it's normal, then the covariance is zero between sample mean and sample standard deviation).

Here is my attempt:

$$Cov[\bar{y}, s] = \mathbb{E}[\bar{y}s]-\mathbb{E}[\bar{y}]\mathbb{E}[s]$$

I already know $\mathbb{E}[\bar{y}]$ and $\mathbb{E}[s]$ but can not figure out $\mathbb{E}[\bar{y}s]$.

Any help would be appreciated!

Note: for some reason my original post was edited and someone changed standard deviation (in original post) to variance. My question is really about the covariance between sample mean and sample STANDARD DEVIATION not sample variance. Thanks!

I am trying to figure out the covariance between sample mean and sample variance from a population. We DO NOT know whether the population is normal (if it's normal, then the covariance is zero between sample mean and sample variance).

Here is my attempt:

$$Cov[\bar{y}, s^2] = \mathbb{E}[\bar{y}s^2]-\mathbb{E}[\bar{y}]\mathbb{E}[s^2]$$

I already know $\mathbb{E}[\bar{y}]$ and $\mathbb{E}[s^2]$ but can not figure out $\mathbb{E}[\bar{y}s^2]$.

Any help would be appreciated!

I am trying to figure out the covariance between sample mean and sample STANDARD DEVIATION (NOT VARIANCE) from a population. We DO NOT know whether the population is normal (if it's normal, then the covariance is zero between sample mean and sample variance).

Here is my attempt:

$$Cov[\bar{y}, s] = \mathbb{E}[\bar{y}s]-\mathbb{E}[\bar{y}]\mathbb{E}[s]$$

I already know $\mathbb{E}[\bar{y}]$ and $\mathbb{E}[s]$ but can not figure out $\mathbb{E}[\bar{y}s]$.

Any help would be appreciated!

Note: for some reason my original post was edited and someone changed standard deviation (in original post) to variance. My question is really about the covariance between sample mean and sample STANDARD DEVIATION not sample variance. Thanks!

I am trying to figure out the covariance between sample mean and sample STANDARD DEVIATION (NOT VARIANCE) from a population. We DO NOT know whether the population is normal (if it's normal, then the covariance is zero between sample mean and sample standard deviation).

Here is my attempt:

$$Cov[\bar{y}, s] = \mathbb{E}[\bar{y}s]-\mathbb{E}[\bar{y}]\mathbb{E}[s]$$

I already know $\mathbb{E}[\bar{y}]$ and $\mathbb{E}[s]$ but can not figure out $\mathbb{E}[\bar{y}s]$.

Any help would be appreciated!

Note: for some reason my original post was edited and someone changed standard deviation (in original post) to variance. My question is really about the covariance between sample mean and sample STANDARD DEVIATION not sample variance. Thanks!