# Questions tagged [expected-value]

The expected value of a random variable is a weighted average of all possible values a random variable can take on, with the weights equal to the probability of taking on that value.

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### Trying to show $E[\hat \beta_1 | \mathbf{X}] = \beta_1$ directly from the definition of $\hat \beta_1$?

Suppose we have the standard simple linear regression model: $$Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i,$$ with $E[\varepsilon_i|X_i] = 0$ and $\text{Var}[\varepsilon_i|X_i] = \sigma^2$. I'm ...
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### Showing that a discrete random variable has the same moments as a Normal Distribution

Suppose I define $X$ to be normally distributed with $\mu = 0, \sigma^2 = 1$, so that $X$ has the pdf $f_{X}(x) = \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2}, \quad -\infty < x <\infty.$ Let discrete ...
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### Interpretation of $|cor(X,Y)|$ as a common variance and generalisation to a bigger number of variables

Assuming that $X, Y$ are standardized random variables, can we interpret value $|E[XY]|$ as a proportion of "common/shared" variance between $X$ and $Y$? If yes, then if $Z$ is a ...
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### How do you write the expected value of an arbitrary random variable $X$ in terms of $F_X$?

The "Darth Vader rule" for the expected value of non-negative random variable is: $$\mathbb{E}(X) = \int \limits_0^\infty (1-F_X(x)) \ dx.$$ This rule applies only to non-negative random ...
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### How can I mathematically prove this time series when $e_t$ has i.i.d distribution?

Not really sure on how to simplify the $y_t$ because there is $y_{t-1}^2$ In order for a time series to be Martingale difference sequence the expected value given all the past value should be 0 and ...
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### Expected Value of $Y/X$ [closed]

Consider two random variable, $X$ and $Y$ such that $E(Y\mid X)=0.5X$ and $E(Y) = 20$ and $E(X) = 10$. Compute $E(Y/X)$.