Statistics: Random Variables and Distributions

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Questions and Answers

Which characteristic is NOT a property of hypergeometric distribution?

  • There are two outcomes in each trial.
  • Sampling is done without replacement.
  • The number of successes in the population is known.
  • The population size is infinite. (correct)

What is the primary reason researchers may avoid using hypergeometric distribution?

  • It requires continuous data.
  • It can only be applied to large populations.
  • The calculations involved can be tedious. (correct)
  • The probabilities must always be equal.

In Poisson distribution, what does the parameter lambda (λ) represent?

  • The average number of occurrences in a given time period. (correct)
  • The maximum number of successes.
  • The total number of occurrences.
  • The range of the possible outcomes.

Which statement about Poisson distribution is correct?

<p>It is applicable for modeling rare events. (A)</p> Signup and view all the answers

What condition must be met for an event to be modeled using Poisson distribution?

<p>The number of occurrences should be consistent per interval. (C)</p> Signup and view all the answers

How is the probability of successes in a hypergeometric distribution calculated?

<p>Using combinations based on the population parameters. (D)</p> Signup and view all the answers

Which example best illustrates a situation suitable for Poisson distribution?

<p>Number of cars passing a traffic light in an hour. (A)</p> Signup and view all the answers

What would prevent the use of Poisson distribution in a given scenario?

<p>A varying average of occurrences during different experiments. (B)</p> Signup and view all the answers

In the hypergeometric distribution, what parameters are essential to define the distribution?

<p>A, N, and n. (C)</p> Signup and view all the answers

What is the sum of all probabilities in Poisson distribution equal to?

<ol> <li>(D)</li> </ol> Signup and view all the answers

What defines a random variable?

<p>A numerical function based on outcomes of a random experiment (A)</p> Signup and view all the answers

Which statement accurately describes discrete random variables?

<p>They can only assume a countably finite number of values. (A)</p> Signup and view all the answers

What is the relationship between the probabilities associated with a random variable?

<p>They always equal one. (B)</p> Signup and view all the answers

In the example of tossing two unbiased coins, how many heads represents X=2?

<p>The outcome HH (B)</p> Signup and view all the answers

What signifies a continuous random variable?

<p>It can assume any value within a given range. (A)</p> Signup and view all the answers

How is the value of a random variable determined?

<p>By the outcomes of a random experiment. (D)</p> Signup and view all the answers

Which of the following is an example of a discrete random variable?

<p>The number of students in a classroom. (D)</p> Signup and view all the answers

Which of the following correctly reflects the term 'sample space' in relation to random variables?

<p>It includes all possible outcomes of a random experiment. (C)</p> Signup and view all the answers

What distinguishes discrete random variables from continuous random variables?

<p>Discrete variables take on a finite number of values, while continuous variables take on an infinite number of values. (A), Discrete variables only include whole numbers, while continuous variables include decimals. (D)</p> Signup and view all the answers

Which option best describes the probability mass function of a discrete random variable?

<p>It is the summary of all probability values for various outcomes. (A)</p> Signup and view all the answers

How is the expected value of a discrete random variable computed?

<p>By summing the products of each outcome and its respective probability. (B)</p> Signup and view all the answers

In the example of throwing two unbiased coins, what is the probability of getting 2 heads?

<p>1/4 (C)</p> Signup and view all the answers

What is the mean or expected value in a discrete distribution represented as?

<p>E(x) = Σ[xP(x)] (B)</p> Signup and view all the answers

What is the significance of the expected value in decision-making?

<p>It helps in evaluating the average outcome over several trials. (C)</p> Signup and view all the answers

In a discrete variable distribution, what does the variance measure?

<p>The spread of the outcomes around the mean. (C)</p> Signup and view all the answers

What is the formula to calculate the variance of a discrete random variable?

<p>σ^2 = Σ(x-m)^2 * P(x) (B)</p> Signup and view all the answers

If X represents the total points from a pair of dice, which sum has the highest probability?

<p>7 (A)</p> Signup and view all the answers

In a discrete probability distribution, the total of all probabilities must equal what?

<p>1 (C)</p> Signup and view all the answers

If the probability of getting 3 power cuts in a day is 0.09, what is the probability of getting no power cuts?

<p>0.37 (D)</p> Signup and view all the answers

When calculating the expected number of power cuts, what does an outcome represent?

<p>A specific number of power cuts on a given day. (B)</p> Signup and view all the answers

When considering the standard deviation of a discrete random variable, what is it defined as?

<p>The square root of the variance. (A)</p> Signup and view all the answers

What is represented by the probability histogram in a discrete distribution?

<p>Displays probability values associated with discrete outcomes. (B)</p> Signup and view all the answers

What is the result of the summation of probabilities in a discrete probability distribution?

<p>It equals 1. (B)</p> Signup and view all the answers

What is the variance in the given example using the standard deviation formula?

<p>1.23 (B)</p> Signup and view all the answers

In a binomial distribution, which of the following conditions must be met?

<p>Each trial must result in two mutually exclusive outcomes. (D)</p> Signup and view all the answers

What does 'p' represent in a binomial distribution?

<p>Probability of success (D)</p> Signup and view all the answers

What is the formula for calculating the probability of x successes in n trials in a binomial distribution?

<p>$P(X = x) = nCx px q^{n-x}$ (A)</p> Signup and view all the answers

Which of the following best describes Bernoulli trials?

<p>Trials with independent and identical conditions with two outcomes. (C)</p> Signup and view all the answers

How does the binomial distribution behave as the number of trials n increases?

<p>It tends to a normal distribution. (C)</p> Signup and view all the answers

What significance does the term 'independence of trials' imply in binomial distribution?

<p>Each trial outcome is independent of others. (A)</p> Signup and view all the answers

In a hypergeometric distribution, what is a key characteristic regarding sampling?

<p>Sampling is performed without replacement. (B)</p> Signup and view all the answers

What is the mean of a binomial distribution expressed as?

<p>np (C)</p> Signup and view all the answers

Which of the following distributions provides a basis for rational decision-making when analyzing data?

<p>Binomial distribution (B)</p> Signup and view all the answers

What does the parameter 'q' represent in the context of binomial distributions?

<p>Probability of failure (A)</p> Signup and view all the answers

What is a defining property of the binomial distribution regarding the sum of probabilities?

<p>The probabilities sum to 1. (B)</p> Signup and view all the answers

What is the main application of hypergeometric distribution in statistics?

<p>Sampling without replacement (A)</p> Signup and view all the answers

Flashcards

Random Variable

A numerical value assigned to the outcome of a random experiment. It can be discrete or continuous.

Discrete Random Variable

A type of random variable that can only take on a finite number of values or a countably infinite number of values.

Continuous Random Variable

A type of random variable that can take on any value within a given range.

Expected Value

The average value of a random variable, weighted by its probability distribution.

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Variance

A measure of the spread or variability of a random variable around its expected value.

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Binomial Distribution

A discrete probability distribution that describes the probability of getting a certain number of successes in a sequence of independent trials.

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Bernoulli Trial

A single trial in a binomial distribution, where there are only two possible outcomes (success or failure).

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Hypergeometric Distribution

A discrete probability distribution that describes the probability of getting a certain number of successes in a sample drawn from a finite population without replacement.

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Standard Deviation

The square root of the variance.

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Expectation

The expected value of a random variable, weighted by its probability distribution.

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Probability of Success (p)

The probability of success in a single Bernoulli trial.

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Probability of Failure (q)

The probability of failure in a single Bernoulli trial.

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Number of Trials (n)

The number of trials in a binomial distribution.

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Number of Successes (x)

The number of successes in a binomial distribution.

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Probability Distribution

It describes how the values of a random variable are distributed among the members of a population.

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Population Mean (µ)

The average value of a random variable, calculated for a large number of observations.

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Sample Mean (xÌ…)

The average value of a random variable, calculated for a sample taken from the population.

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Sample Variance (s²)

The variance of a random variable calculated for a sample.

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Sample Standard Deviation (s)

The standard deviation of a random variable calculated for a sample.

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Discrete Variable

A variable that can only take on a finite number of values or a countably infinite number of values. These values are usually whole numbers and can be counted.

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Continuous Variable

A variable that can take on any value within a given range. These values are usually measured.

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Probability Mass Function (PMF)

A function that assigns probabilities to each possible value of a discrete random variable. It describes the likelihood of each outcome.

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Total Probability Rule

The sum of all probabilities in a probability mass function must equal 1. This represents the fact that one of the possible outcomes must occur.

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Expected Value (E(X))

The expected value is the average value of a discrete random variable, weighted by its probability distribution. It's a long-run average of the variable.

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Variance (σ²)

The variance measures how spread out the values of a discrete random variable are around its expected value. It's a squared measure of variability.

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Standard Deviation (σ)

The standard deviation is the square root of the variance. It gives a measure of the typical deviation of values from the expected value.

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Probability Histogram

A visual representation of a probability mass function where the height of each bar corresponds to the probability of a specific value.

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Probability Polygon

A visual representation of a probability mass function where points are connected by lines. The height of each point corresponds to the probability of a specific value.

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Probability Curve

A smooth curve that approximates the shape of a probability histogram or polygon for a discrete random variable. It represents the probability density function.

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Discrete Variable

The value of a discrete variable cannot be measured but counted. It represents an event or a count of objects. These values are often whole numbers.

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Continuous Variable

The value of a continuous variable is represented by measurements. It can take on any value within a given range, including fractions and decimals.

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Probability Mass Function

A function that assigns probabilities to each possible value of a discrete random variable.

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Population Size (N)

The total number of items in the population from which the sample is drawn.

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Number of Successes in the Population (A)

The total number of successes (e.g., red marbles) in the population.

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Sample Size (n)

Size of the sample you're drawing from the population.

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Number of Successes in the Sample (x)

The exact number of successes (e.g., red marbles) you want to find in your sample.

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Poisson Distribution

A probability distribution that describes rare events happening over a specific time or space.

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Average Rate (λ)

The average number of events expected to occur over the specified time or space.

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Number of Occurrences (x)

The number of times an event occurs within the specified interval, usually represented by a whole number starting from zero.

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Probability of x Occurrences (P(x))

The probability of getting exactly 'x' occurrences over the specified time or space.

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Euler's Number (e)

The base of the natural logarithm, approximately 2.71828.

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Study Notes

Random Variables

  • A random variable is a numerical value associated with the outcome of a random experiment.
  • Its value is related to the sample space.
  • A random variable can be discrete or continuous.

Discrete Random Variables

  • Discrete variables have a finite or countably infinite number of values.
  • Typically represent counts (e.g., number of broken eggs, items purchased, defects).

Continuous Random Variables

  • Continuous variables have an uncountably infinite number of values within a given range.
  • Typically represent measurements (e.g., height, weight, time).

Discrete Probability Distributions

  • A probability mass function (PMF) describes the probability that a discrete random variable X takes on a specific value x.
  • PMF values are between 0 and 1 (inclusive) and the sum of all probabilities equals 1.
  • A discrete probability distribution lists all possible values of a random variable and their corresponding probabilities.

Expected Value (Mean)

  • The expected value (µ) is the mean or average value of a random variable in the long run.
  • Calculated as the sum of each possible value multiplied by its probability.
  • Symbolically: µ = E(x) = Σ [x * P(x)]

Variance

  • Variance (s²) measures the spread or dispersion of a probability distribution around the mean.
  • Calculated by squaring the difference between each value and the mean, multiplying by the probability, and summing the results.
  • Symbolically: s² = Σ [(x - µ)² * P(x)]

Binomial Distribution

  • A discrete probability distribution for the number of successes in a fixed number of independent Bernoulli trials.
  • Two possible outcomes (success or failure) per trial, with constant probability of success (p) in each trial.
  • Parameters: n (number of trials), p (probability of success).
  • Formula: P(X = x) = nCx * px * (1 - p)(n-x)

Bernoulli Trials

  • Repeated independent trials with two possible outcomes (success or failure) and a constant probability of success.
  • A foundation for the binomial distribution.

Additional Binomial Properties

  • The sum of probabilities equals 1.
  • Mean = np; Standard Deviation = √npq

Hypergeometric Distribution

  • A discrete probability distribution for the number of successes in a sample taken without replacement from a finite population.
  • Parameters: N (population size), n (sample size), A (number of successes in the population)..
  • Formula: P(x) = [ (ACx) * (N-ACn-x) ] / (NCn)

Poisson Distribution

  • A discrete probability distribution that describes the probability of a given number of events occurring in a fixed interval of time or space.
  • Parameters: λ (average rate of events).
  • Formula: P(x) = (e-λ * λx) / x!

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