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Let $X_1, \dots, X_n$ be independent normally distributed random variables. What is the distribution of: $$ Y_i = \frac{X_i}{\mathrm{var}(X_1, \dots, X_n)}, $$$$ Y_i = \frac{X_i}{\mathrm{stdDev}(X_1, \dots, X_n)}, $$ where $\mathrm{var}(X_1, \dots, X_n)$$\mathrm{stdDev}(X_1, \dots, X_n)$ is the standard deviation of the sample variance? I came across this in a simulation, where the simulated random variables were "normalised" before being used, but no statistical analysis was provided.

Let $X_1, \dots, X_n$ be independent normally distributed random variables. What is the distribution of: $$ Y_i = \frac{X_i}{\mathrm{var}(X_1, \dots, X_n)}, $$ where $\mathrm{var}(X_1, \dots, X_n)$ is the sample variance? I came across this in a simulation, where the simulated random variables were "normalised" before being used, but no statistical analysis was provided.

Let $X_1, \dots, X_n$ be independent normally distributed random variables. What is the distribution of: $$ Y_i = \frac{X_i}{\mathrm{stdDev}(X_1, \dots, X_n)}, $$ where $\mathrm{stdDev}(X_1, \dots, X_n)$ is the standard deviation of the sample? I came across this in a simulation, where the simulated random variables were "normalised" before being used, but no statistical analysis was provided.

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Distribution of "normalised" Gaussian random variables

Let $X_1, \dots, X_n$ be independent normally distributed random variables. What is the distribution of: $$ Y_i = \frac{X_i}{\mathrm{var}(X_1, \dots, X_n)}, $$ where $\mathrm{var}(X_1, \dots, X_n)$ is the sample variance? I came across this in a simulation, where the simulated random variables were "normalised" before being used, but no statistical analysis was provided.