Defining Random Variables and Probability Distributions - Statistics
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Which bar graph feature indicates an error when graphing a probability distribution for $X$?
Which bar graph feature indicates an error when graphing a probability distribution for $X$?
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Bar heights (probabilities) do not sum to $1$. Valid distributions require all probabilities to sum to exactly 1.
Bar heights (probabilities) do not sum to $1$. Valid distributions require all probabilities to sum to exactly 1.
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What is a random variable in probability?
What is a random variable in probability?
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A function assigning a numerical value to each outcome in a sample space. Maps experimental outcomes to numbers for mathematical analysis.
A function assigning a numerical value to each outcome in a sample space. Maps experimental outcomes to numbers for mathematical analysis.
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Identify all possible values of $X$: roll two dice; $X$ is the sum. What values can $X$ take?
Identify all possible values of $X$: roll two dice; $X$ is the sum. What values can $X$ take?
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$X\in{2,3,4,5,6,7,8,9,10,11,12}$. Minimum sum is 1+1=2, maximum is 6+6=12.
$X\in{2,3,4,5,6,7,8,9,10,11,12}$. Minimum sum is 1+1=2, maximum is 6+6=12.
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What is a random variable in probability?
What is a random variable in probability?
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A function assigning a number to each outcome in a sample space. Maps outcomes to numbers for mathematical analysis.
A function assigning a number to each outcome in a sample space. Maps outcomes to numbers for mathematical analysis.
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What is the sample space for a probability experiment?
What is the sample space for a probability experiment?
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The set of all possible outcomes of the experiment. Contains every result that could occur.
The set of all possible outcomes of the experiment. Contains every result that could occur.
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What is the probability distribution of a discrete random variable $X$?
What is the probability distribution of a discrete random variable $X$?
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A list/table of each $x$ value with its probability $P(X=x)$. Shows how probability is distributed across values.
A list/table of each $x$ value with its probability $P(X=x)$. Shows how probability is distributed across values.
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What two conditions must a discrete distribution satisfy for $X$?
What two conditions must a discrete distribution satisfy for $X$?
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$0\le P(X=x)\le 1$ and $\sum P(X=x)=1$. Each probability is valid and they exhaust all possibilities.
$0\le P(X=x)\le 1$ and $\sum P(X=x)=1$. Each probability is valid and they exhaust all possibilities.
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Which notation represents the probability that $X$ equals $3$?
Which notation represents the probability that $X$ equals $3$?
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$P(X=3)$. Standard notation for probability at a specific value.
$P(X=3)$. Standard notation for probability at a specific value.
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Identify the random variable: flip a coin; let $X$ be the number of heads. What is $X$?
Identify the random variable: flip a coin; let $X$ be the number of heads. What is $X$?
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$X\in{0,1}$. Counts heads: 0 for tails, 1 for heads.
$X\in{0,1}$. Counts heads: 0 for tails, 1 for heads.
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Identify the sample space: roll one fair die once. What is $S$?
Identify the sample space: roll one fair die once. What is $S$?
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$S={1,2,3,4,5,6}$. Die shows 1 through 6 as possible outcomes.
$S={1,2,3,4,5,6}$. Die shows 1 through 6 as possible outcomes.
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Find $P(X=2)$ if $X$ is the outcome of one fair die roll.
Find $P(X=2)$ if $X$ is the outcome of one fair die roll.
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$\frac{1}{6}$. Each outcome equally likely: $\frac{1}{6}$.
$\frac{1}{6}$. Each outcome equally likely: $\frac{1}{6}$.
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Find $P(X=7)$ for two fair dice where $X$ is the sum.
Find $P(X=7)$ for two fair dice where $X$ is the sum.
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$\frac{6}{36}$. Six ways to get 7 out of 36 total outcomes.
$\frac{6}{36}$. Six ways to get 7 out of 36 total outcomes.
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What graph is most appropriate for a discrete probability distribution of $X$?
What graph is most appropriate for a discrete probability distribution of $X$?
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A probability histogram (bar graph) with heights $P(X=x)$. Bars show probabilities for each discrete value.
A probability histogram (bar graph) with heights $P(X=x)$. Bars show probabilities for each discrete value.
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What should the vertical axis represent when graphing a discrete distribution for $X$?
What should the vertical axis represent when graphing a discrete distribution for $X$?
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The probability $P(X=x)$. Heights show likelihood of each outcome.
The probability $P(X=x)$. Heights show likelihood of each outcome.
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What should the horizontal axis represent when graphing a discrete distribution for $X$?
What should the horizontal axis represent when graphing a discrete distribution for $X$?
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The possible values $x$ of the random variable $X$. Shows which outcomes are being measured.
The possible values $x$ of the random variable $X$. Shows which outcomes are being measured.
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Which display is appropriate to show a probability distribution table for $X$?
Which display is appropriate to show a probability distribution table for $X$?
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A two-column table of $x$ and $P(X=x)$. Lists each outcome with its probability.
A two-column table of $x$ and $P(X=x)$. Lists each outcome with its probability.
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Find and correct the error: a distribution lists $P(X=1)=0.6$ and $P(X=2)=0.5$.
Find and correct the error: a distribution lists $P(X=1)=0.6$ and $P(X=2)=0.5$.
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Invalid because $0.6+0.5\ne 1$; probabilities must sum to $1$. Probabilities exceed 1 when summed.
Invalid because $0.6+0.5\ne 1$; probabilities must sum to $1$. Probabilities exceed 1 when summed.
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Find the missing probability: $P(X=0)=0.2$, $P(X=1)=0.5$, $P(X=2)=?$ and totals $1$.
Find the missing probability: $P(X=0)=0.2$, $P(X=1)=0.5$, $P(X=2)=?$ and totals $1$.
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$P(X=2)=0.3$. Use $1-0.2-0.5=0.3$ since probabilities sum to 1.
$P(X=2)=0.3$. Use $1-0.2-0.5=0.3$ since probabilities sum to 1.
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Identify $X$ values: choose a person; $X$ is number of siblings. What type of variable is $X$?
Identify $X$ values: choose a person; $X$ is number of siblings. What type of variable is $X$?
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Discrete random variable. Takes countable values (0, 1, 2, ...).
Discrete random variable. Takes countable values (0, 1, 2, ...).
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Identify the event: roll one die; let $X$ be the outcome. What event corresponds to $X\le 2$?
Identify the event: roll one die; let $X$ be the outcome. What event corresponds to $X\le 2$?
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${1,2}$. Outcomes 1 and 2 satisfy $X\le 2$.
${1,2}$. Outcomes 1 and 2 satisfy $X\le 2$.
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Compute $P(X\le 2)$ for one fair die roll where $X$ is the outcome.
Compute $P(X\le 2)$ for one fair die roll where $X$ is the outcome.
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$\frac{2}{6}$. Two favorable outcomes out of six possible.
$\frac{2}{6}$. Two favorable outcomes out of six possible.
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What is the key difference between a data histogram and a probability histogram for $X$?
What is the key difference between a data histogram and a probability histogram for $X$?
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Data uses frequency; probability uses relative frequency $P(X=x)$. Data shows counts; probability shows proportions.
Data uses frequency; probability uses relative frequency $P(X=x)$. Data shows counts; probability shows proportions.
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What does it mean to define $X$ as a random variable for a quantity of interest?
What does it mean to define $X$ as a random variable for a quantity of interest?
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Assign a number $X( )$ to each outcome $ $ to represent that quantity. Creates a numerical representation of the quantity for each possible outcome.
Assign a number $X( )$ to each outcome $ $ to represent that quantity. Creates a numerical representation of the quantity for each possible outcome.
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What is the standard horizontal axis label when graphing a probability distribution for $X$?
What is the standard horizontal axis label when graphing a probability distribution for $X$?
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Possible values of the random variable $X$. Shows which values the random variable can take.
Possible values of the random variable $X$. Shows which values the random variable can take.
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What is the standard vertical axis label when graphing a probability distribution for $X$?
What is the standard vertical axis label when graphing a probability distribution for $X$?
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Probability $P(X=x)$. Shows the likelihood of each value occurring.
Probability $P(X=x)$. Shows the likelihood of each value occurring.
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What two conditions must a discrete probability distribution satisfy?
What two conditions must a discrete probability distribution satisfy?
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$0\le P(X=x)\le 1$ and $\sum P(X=x)=1$. Probabilities must be between 0 and 1, and sum to exactly 1.
$0\le P(X=x)\le 1$ and $\sum P(X=x)=1$. Probabilities must be between 0 and 1, and sum to exactly 1.
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Identify the random variable: roll a die; $X$ is the number shown. What are possible $X$ values?
Identify the random variable: roll a die; $X$ is the number shown. What are possible $X$ values?
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${1,2,3,4,5,6}$. A standard die shows integers from 1 to 6.
${1,2,3,4,5,6}$. A standard die shows integers from 1 to 6.
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Identify the random variable: flip $2$ coins; $X$ is number of heads. What are possible $X$ values?
Identify the random variable: flip $2$ coins; $X$ is number of heads. What are possible $X$ values?
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${0,1,2}$. Can get 0, 1, or 2 heads when flipping two coins.
${0,1,2}$. Can get 0, 1, or 2 heads when flipping two coins.
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What is $P(X=2)$ if $X$ is the number of heads in $2$ fair coin flips?
What is $P(X=2)$ if $X$ is the number of heads in $2$ fair coin flips?
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$\frac{1}{4}$. Only HH gives 2 heads out of 4 equally likely outcomes: HH, HT, TH, TT.
$\frac{1}{4}$. Only HH gives 2 heads out of 4 equally likely outcomes: HH, HT, TH, TT.
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What is $P(X=1)$ if $X$ is the number of heads in $2$ fair coin flips?
What is $P(X=1)$ if $X$ is the number of heads in $2$ fair coin flips?
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$\frac{1}{2}$. HT and TH each give 1 head, so 2 out of 4 outcomes.
$\frac{1}{2}$. HT and TH each give 1 head, so 2 out of 4 outcomes.
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