16 Questions
What is the formula for an unbiased estimator of the sample variance?
( \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 )
What function in R can be used to estimate the sample variance?
var(x)
What is the purpose of standardization in statistics?
To make the mean and variance of the standardized variable equal to 0 and 1 respectively
What does the formula Z = (X - E[X]) / sqrt(V[X]) represent?
Standardization of X
If the mean of standardized variables is 0, what is typically the value of their variance?
1
The linearity of expectation property allows us to do what with expectations of sums?
Distribute them as products
What is the expected value of the function $log(X)$ when X has the probability distribution given in the text?
1.096542
What property of expectation is used to show that $E[aX + b] = aE[X] + b$?
Linearity of expectation
What is the formula used in the text to estimate the expected value of a random variable X from a sample $(x_1, x_2, ..., x_n)$?
$\frac{1}{n} \sum_{i=1}^n x_i$
In the example given, what is the expected value of the random variable X?
3.5
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
What is the formula for the expected value of a discrete random variable $X$?
$E[X] = ext{Var}[X] = rac{1}{n-1} ext{Var}(X)$
What is the expected value of the number of pips when rolling a fair six-sided die?
3.5
How can you calculate the expected value of a random sample in R?
Using the mean() function
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
Suppose you have a random sample $X_1, X_2, ..., X_n$ with expected value $ar{X}$. What is the best unbiased estimator of $ar{X}$?
$\frac{1}{n} \sum_{i=1}^n X_i$
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)
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.
Make Your Own Quizzes and Flashcards
Convert your notes into interactive study material.
Get started for free
