GATE 2026 EE – Question 64

Engineering Mathematics · Probability and Statistics: Random variables · 2 marks · Numerical answer

Let $X$ and $Y$ be two real-valued random variables with $E(X)=1$, $E(Y)=2$, $E(X^2)=4$, $E(Y^2)=9$, and $E(XY)=0.9$, where $E$ denotes the expectation operator.

The value of $\alpha$ that minimizes $E\left((X-\alpha Y)^2\right)$ is __________.

(Round off to one decimal place)

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Correct answer: 0.09 to 0.11

Explanation

$E[(X-\alpha Y)^2]=E[X^2]-2\alpha E[XY]+\alpha^2E[Y^2]$. Setting the derivative to zero: $-2E[XY]+2\alpha E[Y^2]=0$, so $\alpha=\dfrac{E[XY]}{E[Y^2]}=\dfrac{0.9}{9}=0.1$.