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GATE DA 2024 Q65: Covariance of 'both heads' and 'at least one head'

Question

Two fair coins are tossed independently. XX is a random variable that takes a value of 1 if both tosses are heads and 0 otherwise. YY is a random variable that takes a value of 1 if at least one of the tosses is heads and 0 otherwise.

The value of the covariance of XX and YY is ______ (rounded off to three decimal places).

Try it: numerical answer: type the value

Numerical answer. No negative marking.
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Answer: 0.062 to 0.063 (official GATE 2024 final answer key)

Concept used: Cov(X,Y)=E[XY]−E[X]E[Y]Cov(X, Y) = E[XY] - E[X]E[Y].

First step: E[X]=14E[X] = \tfrac{1}{4}, E[Y]=34E[Y] = \tfrac{3}{4}, and XY=XXY = X because both heads implies at least one head, so E[XY]=14E[XY] = \tfrac{1}{4}.

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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