What is an estimator ?

An estimator is a random variable that is used to estimate a population parameter.

What does it mean if an estimator is unbiased ?

An estimator is called unbiased if its expected value is equal to the population parameter .

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Random Variables (DP IB Applications & Interpretation (AI))

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FrontLinear Combinations of Random Variables
$X$ is a random variable.
Write $E (a X + b)$ in terms of $E (X)$ , where $a$ and $b$ are constants.

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$X$ is a random variable.
Write $E (a X + b)$ in terms of $E (X)$ , where $a$ and $b$ are constants.
If $X$ is a random variable, then $E (a X + b) = a E (X) + b$ , where $a$ and $b$ are constants.
This equation is given in your exam formula booklet.
$X$ is a random variable.
Write $Var (a X + b)$ in terms of $Var (X)$ , where $a$ and $b$ are constants.
If $X$ is a random variable, then $Var (a X + b) = a^{2} Var (X)$ , where $a$ and $b$ are constants.
Adding or subtracting a constant does not affect the variance.
This equation is given in your exam formula booklet.
True or False?
$Var (3 - X) = - Var (X)$ .
False.
Variance can never be negative.
$Var (3 - X) = {(- 1)}^{2} Var (X) = Var (X)$ .
Write $E (a X + b Y)$ in terms of $E (X)$ and $E (Y)$ .
$E (a X + b Y) = a E (X) + b E (Y)$ .
This equation is not given in your exam formula booklet.
Write $Var (a X + b Y)$ in terms of $Var (X)$ and $Var (Y)$ .
$Var (a X + b Y) = a^{2} Var (X) + b^{2} Var (Y)$ .
This equation is not given in your exam formula booklet.
True or False?
$E (a X - b Y) = a E (X) - b E (Y)$ .
True.
$E (a X - b Y) = a E (X) - b E (Y)$ .
True or False?
$Var (a X - b Y) = a^{2} Var (X) - b^{2} Var (Y)$ .
False.
$Var (a X - b Y) = a^{2} VarE (X) + b^{2} Var (Y)$ .
What condition must $X$ and $Y$ satisfy in order for the formula $Var (a X + b Y) = a^{2} Var (X) + b^{2} Var (Y)$ to be true?
$X$ and $Y$ must be independent in order for the formula $Var (a X + b Y) = a^{2} Var (X) + b^{2} Var (Y)$ to be true.
True or False?
$Var (a_{1} X_{1} \pm a_{2} X_{2} \pm . . . \pm a_{n} X_{n}) = a_{1} Var (X_{1}) + a_{2} Var (X_{2}) + . . . + a_{n} Var (X_{n})$ .
False.
$Var (a_{1} X_{1} \pm a_{2} X_{2} \pm . . . \pm a_{n} X_{n}) = {a_{1}}^{2} Var (X_{1}) + {a_{2}}^{2} Var (X_{2}) + . . . + {a_{n}}^{2} Var (X_{n})$ .
This is given in the formula booklet.
You need to square the coefficients of the random variables.
$X_{1}$ and $X_{2}$ are independent observations of a random variable with variance $σ^{2}$ .
Write $Var (X_{1} + X_{2})$ in terms of $σ^{2}$ .
$X_{1}$ and $X_{2}$ are independent observations of a random variable with variance $σ^{2}$ .
$Var (X_{1} + X_{2}) = Var (X_{1}) + Var (X_{2}) = σ^{2} + σ^{2} = 2 σ^{2}$ .
True or False?
If $X_{1}$ and $X_{2}$ are independent observations of the random variable $X$ , then $Var (X_{1} + X_{2}) = Var (2 X)$ .
False.
If $X_{1}$ and $X_{2}$ are independent observations of the random variable $X$ with variance $σ^{2}$ , then $Var (X_{1} + X_{2}) \neq Var (2 X)$ .
$Var (X_{1} + X_{2}) = 2 Var (X)$ and $Var (2 X) = 2^{2} Var (X)$ .
If $X_{1}$ and $X_{2}$ are independent observations of the random variable $X$ , then what is the difference between $X_{1} + X_{2}$ and $2 X$ ?
If $X_{1}$ and $X_{2}$ are independent observations of the random variable $X$ , then :
- $X_{1} + X_{2}$ is where two observations are taken and added together,
- whereas $2 X$ is where one observation is taken and doubled.
True or False?
If $X_{1}$ and $X_{2}$ are observations of the random variable $X$ , then $E (X_{1} + X_{2}) = E (2 X)$ .
True.
If $X_{1}$ and $X_{2}$ are observations of the random variable $X$ with mean $μ$ , then $E (X_{1} + X_{2}) = E (2 X)$ . Both are equal to $2 E (X)$ .
True or False?
If $X ~ Po (m)$ , then $2 X$ also follows a Poisson distribution.
False.
If $X ~ Po (m)$ , then $2 X$ does not follow a Poisson distribution.
$E (2 X) = 2 m$ and $Var (2 X) = 2^{2} m$ , therefore $E (2 X) \neq Var (2 X)$ .
What is an estimator?
An estimator is a random variable that is used to estimate a population parameter.
What does it mean if an estimator is unbiased?
An estimator is called unbiased if its expected value is equal to the population parameter.
True or False?
The sample mean $x = \sum_{} \frac{x}{n}$ is an unbiased estimate for the population mean.
True.
The sample mean $x = \sum_{} \frac{x}{n}$ is an unbiased estimate for the population mean.
True or False?
The variance of a sample ${s_{n}}^{2} = \sum_{} \frac{x^{2}}{n} - {\bar{x}}^{2}$ is an unbiased estimate for the population variance.
False.
The variance of a sample ${s_{n}}^{2} = \sum_{} \frac{x^{2}}{n} - {\bar{x}}^{2}$ is a biased estimate for the population variance.
What is the formula for an unbiased estimate of the population variance?
The formula for an unbiased estimate of the population variance is $s_{n - 1}^{2} = \frac{n}{n - 1} s_{n}^{2}$ .
This formula is given in your exam formula booklet.
True or False?
$s_{n - 1}^{2} = \sum_{} \frac{{(x - \bar{x})}^{2}}{n - 1}$ .
True.
$s_{n - 1}^{2} = \sum_{} \frac{{(x - \bar{x})}^{2}}{n - 1}$ , the formula is similar to the formula for the variance of a sample, except for the denominator.
What is the difference between $s_{n}^{2}$ and $s_{n - 1}^{2}$ , in terms of what they represent?
$s_{n}^{2}$ represents the variance of a sample when it is treated as a population, whereas $s_{n - 1}^{2}$ is an unbiased estimate for the population variance using a sample.

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Random Variables (DP IB Applications & Interpretation (AI))

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Cards in this collection (21)

1. Number & Algebra

Number Toolkit

Exponentials & Logs

Sequences & Series

Financial Applications

Complex Numbers

Further Complex Numbers

Matrices

Eigenvalues & Eigenvectors

2. Functions

Linear Functions & Graphs

Further Functions & Graphs

Modelling with Functions

Functions Toolkit

Transformations of Graphs

Modelling with Logarithmic, Logistic & Piecewise Functions

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

Trigonometric Identities & Equations

Voronoi Diagrams

Matrix Transformations

Vector Properties

Vector Equations of Lines

Modelling with Vectors

Graph Theory

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Non-linear Regression

Probability

Probability Distributions

Random Variables

Binomial Distribution

Normal Distribution

Combinations of Normal Distributions & Sample Mean Distributions

Poisson Distribution

Hypothesis Testing using the Chi-squared Distribution

Hypothesis Testing for Population Parameters

Transition Matrices & Markov Chains

5. Calculus

Differentiation

Further Differentiation

Integration

Further Integration

Kinematics

Differential Equations

Coupled & Second Order Differential Equations