The value of $correlation(X, Y)$ is __________ . (Answer in integer)
We need to find the correlation between random variables $X$ and $Y$, denoted as $Corr(X, Y)$.
Understanding Random Variable Distributions
- $X$ follows a $Uniform(-1, 1)$ distribution.
- The conditional distribution of $Y$ given $X=x$ is $Uniform(x^2 - 0.1, x^2 + 0.1)$.
Calculating Expected Values
First, let's find the expected value of $X$ and $Y$ given $X=x$.
- For $X \sim Uniform(a, b)$, the expected value is $E[X] = \frac{a+b}{2}$.
So, $E[X] = \frac{-1 + 1}{2} = 0$. - For $Y|X=x \sim Uniform(x^2 - 0.1, x^2 + 0.1)$, the conditional expectation is: $E[Y|X=x] = \frac{(x^2 - 0.1) + (x^2 + 0.1)}{2} = \frac{2x^2}{2} = x^2$.
Calculating Overall Expected Value E[Y]
Using the law of total expectation, $E[Y] = E[E[Y|X]]$:
$E[Y] = E[X^2]$
For $X \sim Uniform(a, b)$, $E[X^2] = \frac{a^2 + ab + b^2}{3}$.
Here, $a = -1$ and $b = 1$.
$E[X^2] = \frac{(-1)^2 + (-1)(1) + (1)^2}{3} = \frac{1 - 1 + 1}{3} = \frac{1}{3}$.
Therefore, $E[Y] = \frac{1}{3}$.
Calculating Covariance Cov(X, Y)
The formula for covariance is $Cov(X, Y) = E[XY] - E[X]E[Y]$.
We need $E[XY]$. Using the law of total expectation again:
$E[XY] = E[E[XY|X]] = E[X \cdot E[Y|X]]$
Substitute $E[Y|X] = X^2$:
$E[XY] = E[X \cdot X^2] = E[X^3]$.
For $X \sim Uniform(-1, 1)$, $E[X^k] = \frac{1}{1 - (-1)} \int_{-1}^{1} x^k dx$.
Calculate $E[X^3]$:
$E[X^3] = \frac{1}{2} \int_{-1}^{1} x^3 dx = \frac{1}{2} \left[ \frac{x^4}{4} \right]_{-1}^{1} = \frac{1}{2} \left( \frac{1^4}{4} - \frac{(-1)^4}{4} \right) = \frac{1}{2} \left( \frac{1}{4} - \frac{1}{4} \right) = 0$.
So, $E[XY] = 0$.
Now, calculate the covariance:
$Cov(X, Y) = E[XY] - E[X]E[Y] = 0 - (0) \cdot (\frac{1}{3}) = 0$.
Calculating Correlation Corr(X, Y)
The correlation is calculated as $Corr(X, Y) = \frac{Cov(X, Y)}{\sqrt{Var(X) Var(Y)}}$.
Since $Cov(X, Y) = 0$, and the variances $Var(X)$ and $Var(Y)$ are finite and non-zero (specifically $Var(X) = \frac{(1-(-1))^2}{12} = \frac{4}{12} = \frac{1}{3}$), the correlation is:
$Corr(X, Y) = \frac{0}{\sqrt{Var(X) Var(Y)}} = 0$.
Final Answer
The value of $correlation(X, Y)$ is 0.
Important Questions from Correlation Analysis
- In the variance-covariance matrix given below, which one of the following pairs of elements has the highest correlation?
$$ \begin{bmatrix} & P & Q & R & S \\ P & 0 & 0 & 0 & 0 \\ Q & 0 & 3.73 & 1.85 & 3.23 \\ R & 0 & 1.85 & 0.93 & 1.61 \\ S & 0 & 3.23 & 1.61 & 2.79 \end{bmatrix} $$ - Consider 62 pair of observations $(x_i, y_i)$, i = 1, …., 62, obtained from the distribution of $(X, Y)$. The Pearson correlation coefficient ($R$) of these observations is 0.27.
Which one of the following statements is CORRECT for a significance level $\alpha = 0.05$ and $t_{0.025,60} = 2.0$?
Hint: Test statistic $t = \frac{R\sqrt{n - 2}}{\sqrt{1 - R^2}}$ - Let $X = \cos \theta$ and $Y = \sin \theta$, where $\theta$ is a random variable uniformly distributed over $(0, 2\pi)$. Which of the following options is/are CORRECT?
Hint: $\text{cov}(X, Y) = E[(X - E(X)) (Y - E(Y))]$ - The multiple correlation ranges between :
- Arrange the following correlation coefficients in descending order in terms of their strength of relationship, as defined by common variance :
