Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size 4 from the distribution having pdf f(x) = e^(−x),0 < x < ∞, zero elsewhere. Find P(Y4 >= 3).
QUESTION:
Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size 4 from the distribution having pdf f(x) = e^(−x),0 < x < ∞, zero elsewhere. Find P(Y4 >= 3).
SOLUTION:
Let X and Y be continous random variabels. The join probability density fuction f(x,y) is a function that is non-negative and for which
$\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }f(x,y)dxdy=1$
For every adequate set A the following hold
$P[(X,Y)\in A]=\int \int_{A}^{}f(x,y)dxdy.$
Then, the following is true
$\begin{aligned}P(0<X_{1}<\frac{1}{3},0<X_{2}<\frac{1}{3})&=\int_{0}^{\frac{1}{3}}\int_{0}^{\frac{1}{3}}4x_{1}(1-x_{2}) dx_{1}dx_{2}\\&=\int_{0}^{\frac{1}{3}}4(1-x_{2}) \int_{0}^{\frac{1}{3}}x_{1} dx_{1}dx_{2}\\&=\int_{0}^{\frac{1}{3}}4(1-x_{2}) .(\frac{x_{1}^{2}}{2})|_{0}^{\frac{1}{3}}dx_{2}\\&=\frac{2}{9}\int_{0}^{\frac{1}{3}}(1-x_{2})dx_{2}\\&=\frac{2}{9}(x_{2}-\frac{x_{2}^{2}}{2})|_{0}^{\frac{1}{3}}\\&=\frac{5}{81}\end{aligned}$
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