Developing Empirical Probability Distributions, Expected Value - Statistics
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What is the sample space for $X=\text{number of TV sets per household}$ if observed values are $0$ to $5$?
What is the sample space for $X=\text{number of TV sets per household}$ if observed values are $0$ to $5$?
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${0,1,2,3,4,5}$. Sample space contains all possible values the random variable can take.
${0,1,2,3,4,5}$. Sample space contains all possible values the random variable can take.
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Identify the random variable in: “number of TV sets per household” for a probability distribution.
Identify the random variable in: “number of TV sets per household” for a probability distribution.
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$X=\text{number of TV sets in a randomly selected household}$. Random variable represents the quantity being measured in each trial.
$X=\text{number of TV sets in a randomly selected household}$. Random variable represents the quantity being measured in each trial.
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What is the empirical distribution if counts for $X=0,1,2$ are $5,15,30$ out of $50$?
What is the empirical distribution if counts for $X=0,1,2$ are $5,15,30$ out of $50$?
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$P(0)=0.1,;P(1)=0.3,;P(2)=0.6$. Divide each count by total 50: $\frac{5}{50}=0.1$, $\frac{15}{50}=0.3$, $\frac{30}{50}=0.6$.
$P(0)=0.1,;P(1)=0.3,;P(2)=0.6$. Divide each count by total 50: $\frac{5}{50}=0.1$, $\frac{15}{50}=0.3$, $\frac{30}{50}=0.6$.
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What is $E(X)$ for $X\in{0,1,2}$ with empirical counts $10,30,60$ out of $100$?
What is $E(X)$ for $X\in{0,1,2}$ with empirical counts $10,30,60$ out of $100$?
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$0(0.1)+1(0.3)+2(0.6)=1.5$. Convert counts to probabilities: $\frac{10}{100}=0.1$, $\frac{30}{100}=0.3$, $\frac{60}{100}=0.6$.
$0(0.1)+1(0.3)+2(0.6)=1.5$. Convert counts to probabilities: $\frac{10}{100}=0.1$, $\frac{30}{100}=0.3$, $\frac{60}{100}=0.6$.
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What is the expected number of TV sets in $100$ households if $E(X)=1.7$ sets per household?
What is the expected number of TV sets in $100$ households if $E(X)=1.7$ sets per household?
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$100\cdot 1.7=170$ sets. Multiply expected value per household by number of households.
$100\cdot 1.7=170$ sets. Multiply expected value per household by number of households.
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Identify the missing probability if $P(X=0)=0.1$, $P(X=1)=0.4$, and $P(X=2)=?$ for a distribution.
Identify the missing probability if $P(X=0)=0.1$, $P(X=1)=0.4$, and $P(X=2)=?$ for a distribution.
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$P(X=2)=1-0.1-0.4=0.5$. Probabilities must sum to 1, so missing probability is $1-0.1-0.4$.
$P(X=2)=1-0.1-0.4=0.5$. Probabilities must sum to 1, so missing probability is $1-0.1-0.4$.
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What is the expected count in $n$ trials if the probability of success is $p$?
What is the expected count in $n$ trials if the probability of success is $p$?
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$E(\text{count})=np$. Expected count in $n$ trials equals trials times success probability.
$E(\text{count})=np$. Expected count in $n$ trials equals trials times success probability.
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State the formula for the expected total over $n$ independent observations of $X$ with mean $E(X)$.
State the formula for the expected total over $n$ independent observations of $X$ with mean $E(X)$.
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$E\left(\sum_{i=1}^{n} X_i\right)=nE(X)$. Expected sum of $n$ independent observations is $n$ times single expectation.
$E\left(\sum_{i=1}^{n} X_i\right)=nE(X)$. Expected sum of $n$ independent observations is $n$ times single expectation.
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Find $E(X)$ if $P(X=1)=0.25$, $P(X=2)=0.50$, $P(X=3)=0.25$.
Find $E(X)$ if $P(X=1)=0.25$, $P(X=2)=0.50$, $P(X=3)=0.25$.
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$1(0.25)+2(0.50)+3(0.25)=2$. Weighted average: $1(0.25)+2(0.50)+3(0.25)=0.25+1+0.75$.
$1(0.25)+2(0.50)+3(0.25)=2$. Weighted average: $1(0.25)+2(0.50)+3(0.25)=0.25+1+0.75$.
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Find the expected number in $200$ households if $E(X)=2.3$ TV sets per household.
Find the expected number in $200$ households if $E(X)=2.3$ TV sets per household.
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$200\cdot 2.3=460$ sets. Scale up single household expectation to 200 households.
$200\cdot 2.3=460$ sets. Scale up single household expectation to 200 households.
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What is $E(X)$ if $X\in{0,2}$ with $P(X=0)=0.6$ and $P(X=2)=0.4$?
What is $E(X)$ if $X\in{0,2}$ with $P(X=0)=0.6$ and $P(X=2)=0.4$?
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$0(0.6)+2(0.4)=0.8$. Only two outcomes: $0$ contributes nothing, $2$ contributes $2(0.4)$.
$0(0.6)+2(0.4)=0.8$. Only two outcomes: $0$ contributes nothing, $2$ contributes $2(0.4)$.
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Find and correct the error: “$E(X)=\sum P(X=x)$” for a discrete random variable.
Find and correct the error: “$E(X)=\sum P(X=x)$” for a discrete random variable.
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Correct: $E(X)=\sum x,P(X=x)$. Error: formula missing the $x$ values; must multiply $x$ by $P(X=x)$.
Correct: $E(X)=\sum x,P(X=x)$. Error: formula missing the $x$ values; must multiply $x$ by $P(X=x)$.
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What is $E(X)$ if $X\in{0,1,2,3}$ with probabilities $0.1,0.2,0.3,0.4$?
What is $E(X)$ if $X\in{0,1,2,3}$ with probabilities $0.1,0.2,0.3,0.4$?
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$0(0.1)+1(0.2)+2(0.3)+3(0.4)=2.0$. Sum products: $0+0.2+0.6+1.2=2.0$.
$0(0.1)+1(0.2)+2(0.3)+3(0.4)=2.0$. Sum products: $0+0.2+0.6+1.2=2.0$.
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What is an empirical probability for an outcome with frequency $f$ in $n$ trials?
What is an empirical probability for an outcome with frequency $f$ in $n$ trials?
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$P(\text{outcome})=\frac{f}{n}$. Divides frequency of occurrence by total number of trials.
$P(\text{outcome})=\frac{f}{n}$. Divides frequency of occurrence by total number of trials.
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What is the empirical probability if $18$ of $200$ households have $3$ TVs?
What is the empirical probability if $18$ of $200$ households have $3$ TVs?
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$P(X=3)=0.09$. $\frac{18}{200}=0.09$
$P(X=3)=0.09$. $\frac{18}{200}=0.09$
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What is the empirical probability if $7$ of $50$ households have $0$ TVs?
What is the empirical probability if $7$ of $50$ households have $0$ TVs?
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$P(X=0)=0.14$. $\frac{7}{50}=0.14$
$P(X=0)=0.14$. $\frac{7}{50}=0.14$
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Find the missing probability if $P(0)=0.12$, $P(1)=0.38$, $P(2)=0.27$, $P(3)=0.15$.
Find the missing probability if $P(0)=0.12$, $P(1)=0.38$, $P(2)=0.27$, $P(3)=0.15$.
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$P(4)=0.08$. $1-0.12-0.38-0.27-0.15=0.08$
$P(4)=0.08$. $1-0.12-0.38-0.27-0.15=0.08$
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Identify the error: a “distribution” has $P(0)=0.4$, $P(1)=0.5$, $P(2)=0.3$.
Identify the error: a “distribution” has $P(0)=0.4$, $P(1)=0.5$, $P(2)=0.3$.
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$\sum P(x)=1.2\ne 1$. Probabilities sum to more than 1, violating distribution rules.
$\sum P(x)=1.2\ne 1$. Probabilities sum to more than 1, violating distribution rules.
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What is $E(X)$ for $X\in{0,1,2,3}$ with $P={0.1,0.2,0.3,0.4}$?
What is $E(X)$ for $X\in{0,1,2,3}$ with $P={0.1,0.2,0.3,0.4}$?
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$E(X)=2.0$. $0(0.1)+1(0.2)+2(0.3)+3(0.4)=0+0.2+0.6+1.2=2.0$
$E(X)=2.0$. $0(0.1)+1(0.2)+2(0.3)+3(0.4)=0+0.2+0.6+1.2=2.0$
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What is $E(X)$ for $X\in{1,2,3}$ with $P={0.2,0.5,0.3}$?
What is $E(X)$ for $X\in{1,2,3}$ with $P={0.2,0.5,0.3}$?
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$E(X)=2.1$. $1(0.2)+2(0.5)+3(0.3)=0.2+1.0+0.9=2.1$
$E(X)=2.1$. $1(0.2)+2(0.5)+3(0.3)=0.2+1.0+0.9=2.1$
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What is the expected number in $100$ households if $P(0)=0.2$, $P(1)=0.5$, $P(2)=0.3$?
What is the expected number in $100$ households if $P(0)=0.2$, $P(1)=0.5$, $P(2)=0.3$?
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$110$. $E(X)=0(0.2)+1(0.5)+2(0.3)=1.1$, so $100(1.1)=110$
$110$. $E(X)=0(0.2)+1(0.5)+2(0.3)=1.1$, so $100(1.1)=110$
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What is the expected number in $100$ households if $E(X)=2.3$ TVs per household?
What is the expected number in $100$ households if $E(X)=2.3$ TVs per household?
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$230$. $100 \times 2.3 = 230$
$230$. $100 \times 2.3 = 230$
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What is $E(X)$ if $X$ is always $4$ (that is, $P(X=4)=1$)?
What is $E(X)$ if $X$ is always $4$ (that is, $P(X=4)=1$)?
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$E(X)=4$. When a variable is constant, its expected value equals that constant.
$E(X)=4$. When a variable is constant, its expected value equals that constant.
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What is the sample space for $X$ = number of TV sets per household (discrete, nonnegative)?
What is the sample space for $X$ = number of TV sets per household (discrete, nonnegative)?
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${0,1,2,3,\dots}$. All possible non-negative integer counts.
${0,1,2,3,\dots}$. All possible non-negative integer counts.
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What is the expected count in $n$ trials if the expected value per trial is $E(X)$?
What is the expected count in $n$ trials if the expected value per trial is $E(X)$?
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$\text{Expected total}=n,E(X)$. Multiply expected value per trial by number of trials.
$\text{Expected total}=n,E(X)$. Multiply expected value per trial by number of trials.
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What is $E(X)$ if $P(X=0)=0.1$, $P(X=1)=0.6$, and $P(X=2)=0.3$?
What is $E(X)$ if $P(X=0)=0.1$, $P(X=1)=0.6$, and $P(X=2)=0.3$?
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$E(X)=1.2$. $0(0.1)+1(0.6)+2(0.3)=0+0.6+0.6=1.2$
$E(X)=1.2$. $0(0.1)+1(0.6)+2(0.3)=0+0.6+0.6=1.2$
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What is the expected total in $100$ trials if $E(X)=1.2$ per trial?
What is the expected total in $100$ trials if $E(X)=1.2$ per trial?
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$120$. $100 \times 1.2 = 120$
$120$. $100 \times 1.2 = 120$
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Identify the random variable in: “number of TV sets in a randomly selected household.”
Identify the random variable in: “number of TV sets in a randomly selected household.”
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$X=\text{number of TV sets in one household}$. The variable counts TV sets in a single household.
$X=\text{number of TV sets in one household}$. The variable counts TV sets in a single household.
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What does $E(X)$ represent in context for TVs per household?
What does $E(X)$ represent in context for TVs per household?
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Long-run average TVs per household. The average number of TVs we expect to find per household.
Long-run average TVs per household. The average number of TVs we expect to find per household.
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What is the expected value if a distribution is $P(0)=0.25$, $P(2)=0.75$?
What is the expected value if a distribution is $P(0)=0.25$, $P(2)=0.75$?
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$E(X)=1.5$. $0(0.25)+2(0.75)=0+1.5=1.5$
$E(X)=1.5$. $0(0.25)+2(0.75)=0+1.5=1.5$
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