Random Variables and Probability Distributions - AP Statistics
Card 1 of 30
Define a continuous random variable.
Define a continuous random variable.
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A random variable with an infinite number of possible values in an interval. Can take any value within a range.
A random variable with an infinite number of possible values in an interval. Can take any value within a range.
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Identify the property of a probability distribution function.
Identify the property of a probability distribution function.
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The sum of all probabilities is 1. All probabilities must add to 1.
The sum of all probabilities is 1. All probabilities must add to 1.
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What is the cumulative distribution function (CDF)?
What is the cumulative distribution function (CDF)?
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A function giving the probability that a random variable is less than or equal to a value. Cumulative probability up to a value.
A function giving the probability that a random variable is less than or equal to a value. Cumulative probability up to a value.
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Define the probability mass function (PMF).
Define the probability mass function (PMF).
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A function giving the probability that a discrete random variable is exactly equal to some value. For discrete variables only.
A function giving the probability that a discrete random variable is exactly equal to some value. For discrete variables only.
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What is the probability density function (PDF)?
What is the probability density function (PDF)?
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A function that describes the likelihood of a continuous random variable to take on a value. For continuous variables; area gives probability.
A function that describes the likelihood of a continuous random variable to take on a value. For continuous variables; area gives probability.
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Which property must a PDF satisfy?
Which property must a PDF satisfy?
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The total area under the curve is 1. Area represents total probability.
The total area under the curve is 1. Area represents total probability.
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What is the mean of a probability distribution?
What is the mean of a probability distribution?
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The expected value of the distribution. The central tendency of the distribution.
The expected value of the distribution. The central tendency of the distribution.
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What does it mean if a random variable is normally distributed?
What does it mean if a random variable is normally distributed?
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It follows a bell-shaped curve, symmetric about the mean. 68-95-99.7 rule applies.
It follows a bell-shaped curve, symmetric about the mean. 68-95-99.7 rule applies.
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What is a binomial distribution?
What is a binomial distribution?
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The distribution of the number of successes in a fixed number of independent Bernoulli trials. Fixed trials with constant success probability.
The distribution of the number of successes in a fixed number of independent Bernoulli trials. Fixed trials with constant success probability.
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State the formula for the binomial probability.
State the formula for the binomial probability.
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$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$. Combines combinations with probability.
$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$. Combines combinations with probability.
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What is the hypergeometric distribution?
What is the hypergeometric distribution?
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Distribution of successes in draws without replacement from a finite population. Population size affects probability.
Distribution of successes in draws without replacement from a finite population. Population size affects probability.
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Identify the main difference between binomial and hypergeometric distributions.
Identify the main difference between binomial and hypergeometric distributions.
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Binomial involves replacement; hypergeometric does not. Replacement changes the probability.
Binomial involves replacement; hypergeometric does not. Replacement changes the probability.
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What is the Poisson distribution?
What is the Poisson distribution?
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A distribution representing the number of events in a fixed interval of time or space. Rate parameter $\lambda$ is key.
A distribution representing the number of events in a fixed interval of time or space. Rate parameter $\lambda$ is key.
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What is the main characteristic of a geometric distribution?
What is the main characteristic of a geometric distribution?
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Counts the number of trials until the first success. Memoryless property applies.
Counts the number of trials until the first success. Memoryless property applies.
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Identify the key feature of a uniform distribution.
Identify the key feature of a uniform distribution.
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All outcomes are equally likely within the range. Constant probability density function.
All outcomes are equally likely within the range. Constant probability density function.
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What is the expected value of a uniform distribution over $[a, b]$?
What is the expected value of a uniform distribution over $[a, b]$?
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The midpoint, $\frac{a+b}{2}$. Average of the endpoints.
The midpoint, $\frac{a+b}{2}$. Average of the endpoints.
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Define a Bernoulli random variable.
Define a Bernoulli random variable.
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A random variable with two possible outcomes: success or failure. Simplest discrete distribution.
A random variable with two possible outcomes: success or failure. Simplest discrete distribution.
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What is the probability of success in a Bernoulli trial?
What is the probability of success in a Bernoulli trial?
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Denoted as $p$, where $0 \text{<} p \text{<} 1$. Complement is $(1-p)$.
Denoted as $p$, where $0 \text{<} p \text{<} 1$. Complement is $(1-p)$.
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Find the probability of rolling a 3 on a fair six-sided die.
Find the probability of rolling a 3 on a fair six-sided die.
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$\frac{1}{6}$. Each outcome equally likely.
$\frac{1}{6}$. Each outcome equally likely.
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Determine the probability of drawing a heart from a standard deck of cards.
Determine the probability of drawing a heart from a standard deck of cards.
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$\frac{1}{4}$. 13 hearts in 52 cards.
$\frac{1}{4}$. 13 hearts in 52 cards.
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Find the variance of a fair six-sided die roll.
Find the variance of a fair six-sided die roll.
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$\frac{35}{12}$. $E(X^2) - [E(X)]^2 = \frac{35}{12}$
$\frac{35}{12}$. $E(X^2) - [E(X)]^2 = \frac{35}{12}$
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Identify the type of random variable: number of heads in 10 coin flips.
Identify the type of random variable: number of heads in 10 coin flips.
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Discrete random variable. Countable outcomes (0-10 heads).
Discrete random variable. Countable outcomes (0-10 heads).
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Identify the type of random variable: height of students in a class.
Identify the type of random variable: height of students in a class.
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Continuous random variable. Can take any value in range.
Continuous random variable. Can take any value in range.
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Find the probability of no heads in 3 coin flips.
Find the probability of no heads in 3 coin flips.
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$\frac{1}{8}$. $(\frac{1}{2})^3 = \frac{1}{8}$
$\frac{1}{8}$. $(\frac{1}{2})^3 = \frac{1}{8}$
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Calculate the probability of drawing an ace from a standard deck of cards.
Calculate the probability of drawing an ace from a standard deck of cards.
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$\frac{1}{13}$. 4 aces in 52 cards.
$\frac{1}{13}$. 4 aces in 52 cards.
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What is the probability of getting a sum of 7 when rolling two dice?
What is the probability of getting a sum of 7 when rolling two dice?
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$\frac{1}{6}$. 6 ways to get sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
$\frac{1}{6}$. 6 ways to get sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
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Determine the probability of drawing a red card from a standard deck.
Determine the probability of drawing a red card from a standard deck.
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$\frac{1}{2}$. 26 red cards in 52 total.
$\frac{1}{2}$. 26 red cards in 52 total.
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Find the probability of getting at least one 6 in two dice rolls.
Find the probability of getting at least one 6 in two dice rolls.
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$\frac{11}{36}$. $1 - P(\text{no sixes}) = 1 - (\frac{5}{6})^2$
$\frac{11}{36}$. $1 - P(\text{no sixes}) = 1 - (\frac{5}{6})^2$
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Define a discrete random variable.
Define a discrete random variable.
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A random variable with a countable number of possible values. Can take values like 1, 2, 3, etc.
A random variable with a countable number of possible values. Can take values like 1, 2, 3, etc.
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Identify the type of random variable: number of heads in 10 coin flips.
Identify the type of random variable: number of heads in 10 coin flips.
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Discrete random variable. Countable outcomes (0-10 heads).
Discrete random variable. Countable outcomes (0-10 heads).
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