Finding expected value of E(Y^2)

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If you know the variance of $X$ then you can use the equation,$$ Var(X)=E[X^2]-(E[X])^2$$ to get the value of $E(X^2)$.

But, it's not necessary that you have to get $E(X^2)$ from $E(X)$ only. In general for any random variable $X$ with probability density function $f(x)$ and any measurable function $g(X)$ of $X$, expectation/expected value of $g(X)$ is $$ E[g(X)]=\int_{-\infty}^{\infty} \! g(x)f(x) \, \mathrm{d}x.$$

substituting $g(X)=X^2$ you can get the value of $E(X^2)$. In your case $f(x)=\frac{1}{9.46-1.43}$ for $1.43 \leq x\leq 9.46$ and $f(x)=0$ for other values of $x$ and thus integration domain is from 1.43 to 9.46.

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ArturasJ
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ArturasJ

Updated on August 01, 2022

Comments

  • ArturasJ
    ArturasJ over 1 year

    I have an equation that looks like this:

    $11.16^2 = X^2 + Y^2 \implies 124.5456 - X^2 = Y^2 \implies 124.5456 - E(X^2) = E(Y^2)$ is that correct?

    The X is random variable that is distributed by continuous uniform distribution from 1.43 to 9.46 . Y is also random variable.

    So how could I get $E(X^2)$ from $E(X)$ to find $E(Y^2)$?

    • Daniele A
      Daniele A over 9 years
      Since you know the distribution of $X$ you can simply compute $$ E(X^2)=\int_{-\infty}^{+\infty} x^2 d\mathbb{P}_X = \frac{1}{9.46-1-43}\int_{1.43}^{9.46}x^2 dx $$ where $\mathbb{P}_X$ is the distribution of $X$.