# Tag Info

• 3,884
Accepted

• 31.6k

### $\mathbb{P}(X\le2Y)$ for independent normal distribution

Follow the hint provided by @Henry. Let $X,Y\sim \mathcal{N}(\mu,\sigma^2)$. By using characteristic functions and independence (recall if $X\perp Y$ then $E[f(X)g(Y)]=E[f(X)]E[g(Y)]$ for Borel $f,g$),...
• 8,846

### How do I calculate phi(z) to 4dp when my Normal Distribution tables are to 2dp?

You can get a good approximation with interpolation Your tables presumably give $\Phi(0.31)=0.6217$ though $0.6217195$ is closer $\Phi(0.32)=0.6255$ though $0.6255158$ is closer You want to go $23\%$...
• 143k
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• 12.1k

### (Sheldon Ross) Proving the independence of sample mean and sample variance

It says in the previous sentence or two that $Y$ is independent of the $X_i$, so it is independent of $X_i-\bar X$ (bc $\bar X$ is a function of $X_i$). Since $Y, X_i -\bar X$ has the same ...
• 10.2k
Accepted

### Why is the poisson distribution more symmetrical with a higher mean?

The Poisson distribution takes values on $0,1,2,3,\ldots$. No matter what the mean is, it isn't symmetric because to the right of the mean, it has a tail tending to infinity, while to the left it gets ...
• 81.8k
Accepted

• 468
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• 15.4k

### Distribution and moments of $\frac{X_iX_j}{\sum_{i=1}^n X_i^2}$ when $X_i$'s are i.i.d $N(0,\sigma^2)$

Due to the symmetry $\sigma_1=...=\sigma_n=\sigma$, it is enough to consider $i=1$ and $j=2$. While my first thought was n-dimensional spherical coordinates also, you can immediately do the integrals ...
• 5,393