Question

Suppose we choose independently 25 numbers at random (uniform density) from the interval [0, 20]. Write the normal densities that approximate the densities of their sum S25

Answer 1

Answer:

Step-by-step explanation:

a=0 and b=20 in the uniform density function

∴ the mean μ=$\frac{1}{2}\left ( a+b \right )$=$\frac{1}{2}\left ( 0+20 \right )$=10 and

the variance $\sigma ^{2}=\frac{1}{12}\left ( b-a \right )^{2}$=$\frac{1}{12}\left ( 20-0 ^{2}\right )$=$\frac{100}{3}$.

The standard deviation is the square root of the variance, so $\sigma =\sqrt{\frac{100}{3}}=\frac{10}{\sqrt{3}}$

Having determined the mean and standard deviation of the uniform distribution, we can conclude that $S_{\ 25 }$ follows a normal distribution with $S_{\mu }=n\mu =25\ast 10=250$ and $S_{\\sigma }=\sigma \sqrt{n}=\frac{10}{\sqrt{3}}\ast \sqrt{25}$.

The normal probability distribution is:

$f\left ( x \right )=\frac{1}{\sigma \sqrt{2\pi }}\varrho ^{-\frac{1}{2}}\ast \left ( \frac{x-\mu }{\sigma } \right )^{2}$

So, substituting $S_{\mu }$ and $S_{\sigma }$

$f\left ( x \right )=\frac{\sqrt{3}}{\ 50\sqrt{2\pi }}\varrho ^{-\frac{1}{2}}\ast \left ( \frac{x-250 }{50\sqrt{3}} } \right )^{2}$

 

Having approximated sum $S_{25}$, we move on to the standardized sum $S_{25}\ast$, which is the same $S_{25}$ as only with μ=0 and σ=1. This means the probability distribution $S_{25}\ast$ is the standard normal distribution, which is:

$f\left ( x \right )=\frac{1}{\sigma \sqrt{2\pi }}\varrho ^{\frac{-1}{2}x^{2}}$