Let X and Y be two real-valued random variables with $\text{E}(X) = 1$, $\text{E}(Y) = 2$, $\text{E}(X^2) = 4$, $\text{E}(Y^2) = 9$, and $\text{E}(XY) = 0.9$, where $\text{E}$ denotes the expectation operator. The value of $\alpha$ that minimizes $\text{E}( (X-\alpha Y)^2 )$ is __________ (Round off to one decimal place)
Minimizing Expectation $\text{E}((X-\alpha Y)^2)$
We want to find the value of $\alpha$ that minimizes the function $f(\alpha) = \text{E}( (X-\alpha Y)^2 )$.
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Expand the expression:
$(X-\alpha Y)^2 = X^2 - 2\alpha XY + \alpha^2 Y^2$.
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Apply the expectation operator:
$f(\alpha) = \text{E}(X^2 - 2\alpha XY + \alpha^2 Y^2)$.
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Use linearity of expectation:
$f(\alpha) = \text{E}(X^2) - 2\alpha \text{E}(XY) + \alpha^2 \text{E}(Y^2)$.
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Substitute the given values:
$\text{E}(X^2) = 4$, $\text{E}(Y^2) = 9$, $\text{E}(XY) = 0.9$.
$f(\alpha) = 4 - 2\alpha (0.9) + \alpha^2 (9)$
$f(\alpha) = 9\alpha^2 - 1.8\alpha + 4$.
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Find the minimum using calculus:
To minimize $f(\alpha)$, we take the derivative with respect to $\alpha$ and set it to zero:
$\frac{df}{d\alpha} = \frac{d}{d\alpha}(9\alpha^2 - 1.8\alpha + 4)$
$\frac{df}{d\alpha} = 18\alpha - 1.8$.
Set the derivative to zero:
$18\alpha - 1.8 = 0$.
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Solve for $\alpha$:
$18\alpha = 1.8$
$\alpha = \frac{1.8}{18}$
$\alpha = 0.1$.
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Rounding:
Rounding 0.1 to one decimal place gives 0.1.
The value of $\alpha$ that minimizes $\text{E}( (X-\alpha Y)^2 )$ is 0.1.
Important Questions from Random Variables Basics
The length of time X, needed by an examinee of competition to complete a 1-hour exam, is a random variable with
PDF \(f(x)=\dfrac{6}{5}(x^2+x);0 \le x \le 1.\) , The value of F(0.5) is:If X follows a binomial distribution with n = 6 and \(p=\dfrac{1}{4}\) then the skewness of X is:
If the customers arrive in a shop in Poisson fashion with parameter λ, the fourth raw moment \(\mu_4^{'}\) for the inter-arrival time is:
A discrete random variable X has the probability functions as:
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8
f(x)
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2k
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5k
5k
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The value of E(X) is:What percentage of scores falls within three standard deviations from the mean for the normal variate?
