GATE Mechanical Engineering (ME) 2020 Shift 1 Solved Paper

Question : 47 of 65

Marks: +1, -0

Consider two exponentially distributed random variables X and Y, both having a mean of 0.50. Let Z = X + Y and r be the correlation coefficient between X and Y. If the variance of Z equals 0, then the value of r is ______ (round off to 2 decimal places).

Your Answer:

Validate

Solution:

µ(x)=µ(y)=0.5

∴ σ^{2}(x)=σ^{2}(y)=0.5=0.25

Z=X+Y

Var(z)=var x+var y+2 cov(x, y)

∴ cov(x, y)=\;{-0.25-0.25}/{2}

\;\; =-0.25

∴ r=\;{cov(x, y)}/{σ_{x} σ_{y}}

\;\; =\;{-0.25}/{√{0.25} √{0.25}}

\;\; =-1

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