Solved – Theoretical expected value and variance

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Let $X$ be a random variable having expected value $\mu$ and variance $\sigma^2$. Find the Expected Value and Variance of $Y = \frac{X−\mu}{\sigma}$.

I would like to show some progress I've made so far, but honestly I've been thinking about this problem for the past few days but just have no idea where to start. Any hint or insight on a starting point would be much appreciated.

Thanks!

Best Answer

Given a random variable $X$, a location scale transformation of $X$ is a new random variable $Y=aX+b$ where $a$ and $b$ are constants with $a>0$.

The location scale transformation $aX+b$ horizontally scales the distribution of $X$ by the factor $a$, and then shifts the distribution so obtained by the factor $b$ on the real line $\mathbb{R}$.

  • In an intuitive sense, the expected value $\mathbb{E}[X]$ of a random variable is the center of mass of the distribution of $X$. Shifting the distribution of $X$ by a factor $b$, shifts the center of mass by the factor $b$. Scaling the distribution of $X$ by a factor $a$, scales the center of mass by $a$. In other words, $$\mathbb{E}[aX+b]=a\mathbb{E}[X]+b$$
  • Similarly, the variance of $X$ is a measure of the horizontal spread of the distribution of $X$, but the $\text{Var}[X]$ is defined as squared-distance. Thus scaling the distribution of $X$ by a factor $a$, scales the $\text{Var}[X]$ by the factor $a^2$. Shifting the distribution of $X$ by any factor will not affect the spread of distribution, $\text{Var}[X]$, but only affects center of mass. In other words, $$\text{Var}[aX+b]=\text{Var}[aX]=a^2\text{Var}[X]$$

Now, here is an hint to your problem: $Y=\dfrac{X-\mu}{\sigma}=\dfrac{1}{\sigma}X-\dfrac{\mu}{\sigma}$, which can be written as $aX+b$. Find $a$ and $b$, and then use the location-scale transformation.

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