Let Z be an exponential random variable with mean 1 . That is, the cumulative distribution function of Z is given by F_{z}(x)=\\begin{array}{c}1-e^{-x\end{array} , \text{ if } x \geq 0 ; 0 , \text{ if } x 2 ∣ Z>1), rounded off to two decimal places, is equal to

GATE Electronics and Communications (EC) 2019 Solved Paper

Question : 32 of 65

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Let

Z

be an exponential random variable with mean 1 . That is, the cumulative distribution function of

Z

is given by

F_{z}(x)=\{\table 1-e^{-x} , \;\text " if "\; x ≥ 0 ; 0 , \;\text " if "\; x<0.

Then

Pr(Z>2 ∣ Z>1)

, rounded off to two decimal places, is equal to

Solution:

Probability function

( {Pdf})=\;{ {d}}/{ {dx}}( {CDF})

∴ P d f=\{\table e^{-x}, , x ≥ 0 ; 0 , , \;\; x<0.

Now

Pr(z>2 ∣ z>1)=\;{Pr[(z>2) ∩(z>1)]}/{Pr(z>1)}

=\;{Pr(z>2)}/{Pr(z>1)}

=\;{∫↙{2}↖{∞} e^{-x} d x}/{∫↙{1}↖{∞} e^{-x} d x}

=\;{-1(e^{-∞}-e^{-2})}/{-1(e^{-∞}-e^{-1})}

=\;{ {e}^{-2}}/{ {e}^{-1}}=\;{1}/{ {e}}

∴ Pr( {Z}>2 ∣ {z}>1)=0.367

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