Respuesta :
Answer:
Step-by-step explanation:
a=0 and b=20 in the uniform density function
∴ the mean μ=[tex]\frac{1}{2}\left ( a+b \right )[/tex]=[tex]\frac{1}{2}\left ( 0+20 \right )[/tex]=10 and
the variance [tex]\sigma ^{2}=\frac{1}{12}\left ( b-a \right )^{2}[/tex]=[tex]\frac{1}{12}\left ( 20-0 ^{2}\right )[/tex]=[tex]\frac{100}{3}\\[/tex].
The standard deviation is the square root of the variance, so [tex]\sigma =\sqrt{\frac{100}{3}}=\frac{10}{\sqrt{3}}[/tex]
Having determined the mean and standard deviation of the uniform distribution, we can conclude that [tex]S_{\ 25 }[/tex] follows a normal distribution with [tex]S_{\mu }=n\mu =25\ast 10=250[/tex] and [tex]S_{\\sigma }=\sigma \sqrt{n}=\frac{10}{\sqrt{3}}\ast \sqrt{25}[/tex].
The normal probability distribution is:
[tex]f\left ( x \right )=\frac{1}{\sigma \sqrt{2\pi }}\varrho ^{-\frac{1}{2}}\ast \left ( \frac{x-\mu }{\sigma } \right )^{2}[/tex]
So, substituting [tex]S_{\mu }[/tex] and [tex]S_{\sigma }[/tex]
[tex]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}[/tex]
Having approximated sum [tex]S_{25}[/tex], we move on to the standardized sum [tex]S_{25}\ast[/tex], which is the same [tex]S_{25}[/tex] as only with μ=0 and σ=1. This means the probability distribution [tex]S_{25}\ast[/tex] is the standard normal distribution, which is:
[tex]f\left ( x \right )=\frac{1}{\sigma \sqrt{2\pi }}\varrho ^{\frac{-1}{2}x^{2}}[/tex]
