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 .

IBMathsDPApplications & Interpretation (AI)HLFlashcards4. Statistics & ProbabilityRandom Variables

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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Cards in this collection (21)

$X$ 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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