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Let $X_1$, $X_2$, $X_3$ be random variable having same density function: $f(x)=2x/9$ when $0 < x < 3$ and $f(x)=0$ elsewhere.
Let $Y=X_1+X_2+X_3$.
I need to find the mean and variance of $Y$.
I know how to calculate mean, but for the variance do I need covariance of $X_1$, $X_2$ and $X_3$?
How can I solve for them?

  • 0
    Does you assignment state that $X_1$, $X_2$ and $X_3$ are independent?2012-05-02
  • 0
    No, that's the problem. There is no more information about random variables.2012-05-02
  • 0
    Then there's not enough information to solve the problem. You might be able to find bounds for the variance, though.2012-05-02
  • 1
    The minimum possible variance is $0$. This would be the case, for example, if $X_1 = \sqrt{2/3} (Y_1 - Y_2/2 - Y_3/2)$, $X_2 = \sqrt{2/3} (-Y_1/2 +Y_2 - Y_3/2)$, $X_3 = \sqrt{2/3} (-Y_1/2 - Y_2/2 + Y_3)$ where $Y_1$, $Y_2$, $Y_3$ are independent random variables with the given distribution. The maximum possible variance is $9$ times the variance of $X_1$, obtained when $X_1, X_2, X_3$ are all the same random variable.2012-05-02

3 Answers 3