Question

Suppose X has a continuous uniform distribution over the interval Round your answers to 3 decimal places. (a) Determine the mean, variance, and standard deviation of X 1-1, 」 Mean- 0.33 Variance- Standard deviation-(0.574 (b) Determine the value for x such that 0.82

Answer 1

Solution :

$\mathrm{Let\:\:X\:\:be\:a\:random\:variable\:with\:continuous\:uniform\:distribution\:over\:the\:interval\:\left[-1,\:1\right]\:with\:the\:pdf\::}$

$f(x)=\frac{1}{1-(-1)}=\frac{1}{2},\:\:-1<x<1$

( a )

$\mathrm{Mean\:\:E(X)}=\frac{-1&plus;1}{2}=0$

$\mathrm{Var(X)}=\frac{(1-(-1))^{2}}{12}=\frac{(2)^{2}}{12}=\frac{4}{12}=\frac{1}{3}=0.33333\approx 0.333$

$\mathrm{Std(X)}=\sqrt{\left ( \frac{1}{3} \right )}=0.57735\approx 0.577$

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( b )

$\mathrm{P}(-x<\mathrm{X}<x)=\int_{-x}^{x}\frac{1}{2}dx=\frac{1}{2}*2x=x=0.82$