Mean and Standard Deviation - AP Statistics
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What is the variance of a uniformly distributed random variable $X$ over $a$ to $b$?
What is the variance of a uniformly distributed random variable $X$ over $a$ to $b$?
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$Var(X) = \frac{(b-a)^2}{12}$. Uniform variance uses range squared divided by 12.
$Var(X) = \frac{(b-a)^2}{12}$. Uniform variance uses range squared divided by 12.
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Identify the mean of a uniformly distributed random variable $X$ over $a$ to $b$.
Identify the mean of a uniformly distributed random variable $X$ over $a$ to $b$.
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$E(X) = \frac{a+b}{2}$. Uniform mean is the midpoint of the interval.
$E(X) = \frac{a+b}{2}$. Uniform mean is the midpoint of the interval.
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What is the standard deviation of a binomial random variable $X$ with $n$ trials and probability $p$?
What is the standard deviation of a binomial random variable $X$ with $n$ trials and probability $p$?
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$\text{SD}(X) = \text{sqrt}(n \times p \times (1-p))$. Binomial standard deviation uses $n$, $p$, and $(1-p)$ under square root.
$\text{SD}(X) = \text{sqrt}(n \times p \times (1-p))$. Binomial standard deviation uses $n$, $p$, and $(1-p)$ under square root.
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What is the mean of a binomial random variable $X$ with $n$ trials and probability $p$?
What is the mean of a binomial random variable $X$ with $n$ trials and probability $p$?
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$E(X) = n \times p$. Binomial mean equals number of trials times success probability.
$E(X) = n \times p$. Binomial mean equals number of trials times success probability.
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How do you calculate the standard deviation from variance?
How do you calculate the standard deviation from variance?
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Take the square root of the variance, $\text{SD}(X) = \text{sqrt}(Var(X))$. Standard deviation is always the positive square root of variance.
Take the square root of the variance, $\text{SD}(X) = \text{sqrt}(Var(X))$. Standard deviation is always the positive square root of variance.
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State the formula for the variance of a discrete random variable $X$.
State the formula for the variance of a discrete random variable $X$.
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$Var(X) = \text{sum of } [(x_i - \bar{x})^2 \times p_i]$. Each squared deviation from mean times its probability, then sum.
$Var(X) = \text{sum of } [(x_i - \bar{x})^2 \times p_i]$. Each squared deviation from mean times its probability, then sum.
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What is the formula for the expected value of a discrete random variable $X$?
What is the formula for the expected value of a discrete random variable $X$?
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$E(X) = \text{sum of } (x_i \times p_i)$. Each value times its probability, then sum all products.
$E(X) = \text{sum of } (x_i \times p_i)$. Each value times its probability, then sum all products.
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Find the mean of a random variable $X$ with $E(X)=4$ after $Y=2X-3$.
Find the mean of a random variable $X$ with $E(X)=4$ after $Y=2X-3$.
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$E(Y) = 5$. Apply transformation: $E(Y) = 2(4) - 3 = 5$.
$E(Y) = 5$. Apply transformation: $E(Y) = 2(4) - 3 = 5$.
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Find the variance of a random variable $X$ with $Var(X)=9$ after $Y=2X-3$.
Find the variance of a random variable $X$ with $Var(X)=9$ after $Y=2X-3$.
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$Var(Y) = 36$. Apply variance rule: $Var(Y) = 2^2 \times 9 = 36$.
$Var(Y) = 36$. Apply variance rule: $Var(Y) = 2^2 \times 9 = 36$.
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What is the mean of $X$ if $X$ follows a normal distribution with mean $\text{mu}$?
What is the mean of $X$ if $X$ follows a normal distribution with mean $\text{mu}$?
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$E(X) = \text{mu}$. Normal distribution mean is the location parameter $\mu$.
$E(X) = \text{mu}$. Normal distribution mean is the location parameter $\mu$.
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What is the variance of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$?
What is the variance of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$?
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$Var(X) = \text{sigma}^2$. Normal distribution variance is $\sigma^2$ by definition.
$Var(X) = \text{sigma}^2$. Normal distribution variance is $\sigma^2$ by definition.
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Identify the standard deviation of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$.
Identify the standard deviation of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$.
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Standard deviation is $\text{sigma}$. Standard deviation is square root of variance: $\sqrt{\sigma^2} = \sigma$.
Standard deviation is $\text{sigma}$. Standard deviation is square root of variance: $\sqrt{\sigma^2} = \sigma$.
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Calculate the mean of $X$ with $E(X)=3$ after $Y=-X$.
Calculate the mean of $X$ with $E(X)=3$ after $Y=-X$.
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$E(Y) = -3$. Multiplying by $-1$ changes sign: $E(-X) = -E(X)$.
$E(Y) = -3$. Multiplying by $-1$ changes sign: $E(-X) = -E(X)$.
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Calculate the variance of $X$ with $Var(X)=16$ after $Y=-X$.
Calculate the variance of $X$ with $Var(X)=16$ after $Y=-X$.
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$Var(Y) = 16$. Multiplying by $-1$ doesn't change variance: $(-1)^2 = 1$.
$Var(Y) = 16$. Multiplying by $-1$ doesn't change variance: $(-1)^2 = 1$.
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For $X$ uniform from 2 to 8, calculate the variance.
For $X$ uniform from 2 to 8, calculate the variance.
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Variance is $3$. Uniform variance: $\frac{(8-2)^2}{12} = \frac{36}{12} = 3$.
Variance is $3$. Uniform variance: $\frac{(8-2)^2}{12} = \frac{36}{12} = 3$.
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What is the mean of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?
What is the mean of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?
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$E(X) = \frac{1}{\text{lambda}}$. Exponential mean is reciprocal of rate parameter.
$E(X) = \frac{1}{\text{lambda}}$. Exponential mean is reciprocal of rate parameter.
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What is the variance of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?
What is the variance of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?
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$Var(X) = \frac{1}{\text{lambda}^2}$. Exponential variance is reciprocal of rate squared.
$Var(X) = \frac{1}{\text{lambda}^2}$. Exponential variance is reciprocal of rate squared.
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If $X$ is a random variable with mean 10, what is $E(2X + 3)$?
If $X$ is a random variable with mean 10, what is $E(2X + 3)$?
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$E(2X + 3) = 23$. Apply linear transformation: $E(2X + 3) = 2(10) + 3$.
$E(2X + 3) = 23$. Apply linear transformation: $E(2X + 3) = 2(10) + 3$.
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If $X$ is a random variable with variance 25, what is $Var(2X + 3)$?
If $X$ is a random variable with variance 25, what is $Var(2X + 3)$?
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$Var(2X + 3) = 100$. Constant doesn't affect variance: $Var(2X + 3) = 4(25)$.
$Var(2X + 3) = 100$. Constant doesn't affect variance: $Var(2X + 3) = 4(25)$.
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What is the variance of $X$ if $X$ is a geometric random variable with probability $p$?
What is the variance of $X$ if $X$ is a geometric random variable with probability $p$?
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$Var(X) = \frac{1-p}{p^2}$. Geometric variance uses $(1-p)$ in numerator, $p^2$ in denominator.
$Var(X) = \frac{1-p}{p^2}$. Geometric variance uses $(1-p)$ in numerator, $p^2$ in denominator.
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What is the mean of $X$ if $X$ is a geometric random variable with probability $p$?
What is the mean of $X$ if $X$ is a geometric random variable with probability $p$?
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$E(X) = \frac{1}{p}$. Geometric mean is reciprocal of success probability.
$E(X) = \frac{1}{p}$. Geometric mean is reciprocal of success probability.
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Calculate the variance for $X$ with $Var(X) = 4$ after transformation $Y = 3X + 2$.
Calculate the variance for $X$ with $Var(X) = 4$ after transformation $Y = 3X + 2$.
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$Var(Y) = 36$. Apply variance rule: $Var(Y) = 3^2 \times 4 = 36$.
$Var(Y) = 36$. Apply variance rule: $Var(Y) = 3^2 \times 4 = 36$.
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Calculate the mean of a random variable $X$ with $E(X) = 5$ after transformation $Y = 3X + 2$.
Calculate the mean of a random variable $X$ with $E(X) = 5$ after transformation $Y = 3X + 2$.
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$E(Y) = 17$. Apply linear transformation: $E(Y) = 3(5) + 2 = 17$.
$E(Y) = 17$. Apply linear transformation: $E(Y) = 3(5) + 2 = 17$.
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What is the variance of a Poisson random variable with rate $\text{lambda}$?
What is the variance of a Poisson random variable with rate $\text{lambda}$?
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$Var(X) = \text{lambda}$. For Poisson distribution, variance equals the rate parameter.
$Var(X) = \text{lambda}$. For Poisson distribution, variance equals the rate parameter.
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What is the mean of a Poisson random variable with rate $\text{lambda}$?
What is the mean of a Poisson random variable with rate $\text{lambda}$?
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$E(X) = \text{lambda}$. For Poisson distribution, mean equals the rate parameter.
$E(X) = \text{lambda}$. For Poisson distribution, mean equals the rate parameter.
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For a random variable $X$, what happens to the standard deviation if $X$ is multiplied by $b$?
For a random variable $X$, what happens to the standard deviation if $X$ is multiplied by $b$?
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Standard deviation is multiplied by $|b|$. Multiplying by $b$ scales standard deviation by $|b|$.
Standard deviation is multiplied by $|b|$. Multiplying by $b$ scales standard deviation by $|b|$.
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Calculate the mean of $X$ if $X$ is uniform from 1 to 5.
Calculate the mean of $X$ if $X$ is uniform from 1 to 5.
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$E(X) = 3$. Uniform from 1 to 5: mean = $(1+5)/2 = 3$.
$E(X) = 3$. Uniform from 1 to 5: mean = $(1+5)/2 = 3$.
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What is the effect on the variance if $X$ is shifted by $c$, i.e., $Y = X + c$?
What is the effect on the variance if $X$ is shifted by $c$, i.e., $Y = X + c$?
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Variance remains unchanged, $Var(Y) = Var(X)$. Adding constant doesn't affect variability measures.
Variance remains unchanged, $Var(Y) = Var(X)$. Adding constant doesn't affect variability measures.
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What is the effect on the mean if $X$ is shifted by $c$, i.e., $Y = X + c$?
What is the effect on the mean if $X$ is shifted by $c$, i.e., $Y = X + c$?
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Mean increases by $c$, $E(Y) = E(X) + c$. Adding constant shifts mean but doesn't change spread.
Mean increases by $c$, $E(Y) = E(X) + c$. Adding constant shifts mean but doesn't change spread.
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Find the standard deviation for $Var(X)=4$.
Find the standard deviation for $Var(X)=4$.
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$\text{SD}(X) = 2$. Standard deviation is the square root of variance.
$\text{SD}(X) = 2$. Standard deviation is the square root of variance.
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