Expectation Quiz: Test Your Knowledge of Expected Value

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Questions and Answers

What is the formula for an unbiased estimator of the sample variance?

$ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 $

$ \frac{1}{n-2} \sum_{i=1}^{n} (x_i - \bar{x})^2 $

$ \frac{1}{n+1} \sum_{i=1}^{n} (x_i - \bar{x})^2 $

$ \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 $ (correct)

What function in R can be used to estimate the sample variance?

sqrt(var(x))

mean(x)

sd(x)

var(x) (correct)

What is the purpose of standardization in statistics?

To increase the variability of the data

To simplify calculations

To make the mean and variance of the standardized variable equal to 0 and 1 respectively (correct)

To introduce bias into the data

What does the formula Z = (X - E[X]) / sqrt(V[X]) represent?

Standardization of X Signup and view all the answers

If the mean of standardized variables is 0, what is typically the value of their variance?

1 Signup and view all the answers

The linearity of expectation property allows us to do what with expectations of sums?

Distribute them as products Signup and view all the answers

What is the expected value of the function $log(X)$ when X has the probability distribution given in the text?

1.096542 Signup and view all the answers

What property of expectation is used to show that $E[aX + b] = aE[X] + b$?

Linearity of expectation Signup and view all the answers

What is the formula used in the text to estimate the expected value of a random variable X from a sample $(x_1, x_2, ..., x_n)$?

$\frac{1}{n} \sum_{i=1}^n x_i$ Signup and view all the answers

In the example given, what is the expected value of the random variable X?

3.5 Signup and view all the answers

What is the purpose of the weighted sum formula $sum(pips * probability)$ in the context of the text?

To calculate the expected value of the random variable X Signup and view all the answers

What is the formula for the expected value of a discrete random variable $X$?

$E[X] = ext{Var}[X] = rac{1}{n-1} ext{Var}(X)$ Signup and view all the answers

What is the expected value of the number of pips when rolling a fair six-sided die?

3.5 Signup and view all the answers

How can you calculate the expected value of a random sample in R?

Using the <code>mean()</code> function Signup and view all the answers

Which of the following properties of expected value is correct?

The expected value of a sum of random variables is the sum of their expected values Signup and view all the answers

Suppose you have a random sample $X_1, X_2, ..., X_n$ with expected value $ar{X}$. What is the best unbiased estimator of $ar{X}$?

$\frac{1}{n} \sum_{i=1}^n X_i$ Signup and view all the answers

Study Notes

Expected Value of a Discrete Random Variable

The expected value of a discrete random variable X is the sum of all possible x_i, where each possible x_i is multiplied by the probability of its occurrence: E[X] = ∑x_i * f(x_i)
Example: fair dice, X is the number of points (pips) on top, possible outcomes are {1, 2, 3, 4, 5, 6} with f(x_i) = 1/6 ∀ i ∈ {1, 2,..., 6}

Estimation of the Expected Value

The expected value is the first moment of a theoretical distribution
When working with data, the expected value must be estimated, typically done with the arithmetic mean: x̄ = (1/n) * ∑x_i
In R, the arithmetic mean of a sample vector x is calculated with mean(x)

Linearity of Expectation

E[aX + b] = aE[X] + b
Proof: E[aX + b] = ∑(ax_i + b) * f(x_i) = aE[X] + b

Variance and Standard Deviation

An unbiased estimator for a sample variance is (1/(n-1)) * ∑(x_i - x̄)^2
In R, the sample variance is calculated with var(x) and the standard deviation with sd(x) or sqrt(var(x))

Standardization of Random Variables

A random variable X can be standardized by subtracting the mean and dividing by the standard deviation: Z = (X - E[X]) / √V[X]
Standardization with sample moments: age_standardized = (age - mean(age)) / sd(age)

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Description

Test your understanding of calculating expected values and applying the linearity of expectation property in probability theory. Includes examples involving functions of random variables and proofs using mathematical expressions.

Expected Value and Linearity of Expectation Quiz

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