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IBMathsDPApplications & Interpretation (AI)HLFlashcards4. Statistics & ProbabilityProbability Distributions

Probability Distributions (DP IB Applications & Interpretation (AI))

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  • What is a discrete random variable?

    A discrete random variable is a variable that can only take certain values within a set.

    Often this involves counting something (for example, the number of heads when a coin is tossed 10 times).

  • What is a probability distribution?

    A probability distribution is a description of all possible values a random variable can take, along with their associated probabilities.

  • True or False?

    Random variables are denoted using lower case letters.

    False.

    Random variables are denoted using upper case letters.

  • What does the notation straight P open parentheses X equals x close parentheses mean?

    straight P open parentheses X equals x close parentheses is the probability that the random variable X takes the value x.

    E.g. straight P open parentheses X equals 3 close parentheses is the probability that the random variable X takes the value 3.

  • What is a discrete uniform distribution?

    A discrete uniform distribution is a probability distribution where a random variable takes a finite number of values, and each value has an equal probability of occurring.

  • What is the probability of each value occurring in a discrete uniform distribution with n values?

    The probability of each value occurring in a discrete uniform distribution with n values is 1 over n.

  • What is the sum of probabilities for all values of a discrete random variable?

    The sum of probabilities for all values of a discrete random variable is 1.

  • True or False?

    straight P open parentheses X greater than k close parentheses equals 1 minus straight P open parentheses X less or equal than k close parentheses

    True.

    straight P open parentheses X greater than k close parentheses equals 1 minus straight P open parentheses X less or equal than k close parentheses

  • True or False?

    The probability that random variable X is at least equal to k can be written as straight P open parentheses X less or equal than k close parentheses.

    False.

    The probability that random variable X is at least equal to k is written as straight P open parentheses X greater or equal than k close parentheses.

    straight P open parentheses X less or equal than k close parentheses is the probability that X is at most equal to k, or is less than or equal to k .

  • True or False?

    A discrete random variable can only take a finite number of values.

    False.

    A discrete random variable can take a finite or infinite number of values.

  • Define expected value.

    Expected value is the average outcome of an experiment if it is repeated many times.

    It is calculated as the sum of each possible outcome multiplied by its probability.

  • What is straight E open parentheses X close parentheses the notation for?

    straight E open parentheses X close parentheses is the expected value (or the mean) of a random variable X.

  • True or False?

    The expected value of a random variable is always a whole number.

    False.

    The expected value of a random variable can be any real number, including fractions or decimals.

  • True or False?

    The expected value straight E open parentheses X close parentheses must be an obtainable value of X.

    False.

    The expected value straight E open parentheses X close parentheses does not need to be an obtainable value of X.

    For example, the expected value when rolling a fair 6-sided dice is 3.5, even though only the numbers 1, 2, 3, 4, 5 and 6 appear on the dice.

  • What is the formula for calculating straight E open parentheses X close parentheses for a discrete random variable?

    The formula for calculating straight E open parentheses X close parentheses for a discrete random variable is straight E open parentheses X close parentheses equals sum x straight P open parentheses X equals x close parentheses

    Where:

    • x is a value that the random variable X can take

    • straight P open parentheses X equals x close parentheses is the probability that X takes the value x

    • sum means to sum up the products x straight P open parentheses X equals x close parentheses for all possible values x

    This equation is in the exam formula booklet.

  • What is a fair game in terms of expected value?

    In terms of expected value, a fair game is one where straight E open parentheses X close parentheses equals 0.

  • True or False?

    When used with reference to the outcome of a game, a positive value for straight E open parentheses X close parentheses means the player can expect to make a loss.

    False.

    When used with reference to the outcome of a game, a positive value for straight E open parentheses X close parentheses means the player can expect to make a gain.

    A negative value for straight E open parentheses X close parentheses means the player can expect to make a loss.

  • How is the expected gain or loss typically calculated in a game?

    The expected gain or loss for a game is typically calculated by subtracting the cost to play the game from the expected value of the prize.

  • Define symmetrical distribution.

    A symmetrical distribution is one where the possible values of X are symmetrical about a middle value, and their probabilities are also symmetrical.

  • What is special about the mean of a symmetrical distribution?

    The mean of a symmetrical distribution is the same as its median.

Previous:ProbabilityNext:Random Variables

    1. Topic 1

      Number Toolkit

    2. Topic 2

      Exponentials & Logs

    3. Topic 3

      Sequences & Series

    4. Topic 4

      Financial Applications

    5. Topic 5

      Complex Numbers

    6. Topic 6

      Further Complex Numbers

    7. Topic 7

      Matrices

    8. Topic 8

      Eigenvalues & Eigenvectors

    1. Topic 1

      Linear Functions & Graphs

    2. Topic 2

      Further Functions & Graphs

    3. Topic 3

      Modelling with Functions

    4. Topic 4

      Functions Toolkit

    5. Topic 5

      Transformations of Graphs

    6. Topic 6

      Modelling with Logarithmic, Logistic & Piecewise Functions

    1. Topic 1

      Geometry Toolkit

    2. Topic 2

      Geometry of 3D Shapes

    3. Topic 3

      Trigonometry

    4. Topic 4

      Trigonometric Identities & Equations

    5. Topic 5

      Voronoi Diagrams

    6. Topic 6

      Matrix Transformations

    7. Topic 7

      Vector Properties

    8. Topic 8

      Vector Equations of Lines

    9. Topic 9

      Modelling with Vectors

    10. Topic 10

      Graph Theory

    1. Topic 1

      Statistics Toolkit

    2. Topic 2

      Correlation & Regression

    3. Topic 3

      Non-linear Regression

    4. Topic 4

      Probability

    5. Topic 5

      Probability Distributions

    6. Topic 6

      Random Variables

    7. Topic 7

      Binomial Distribution

    8. Topic 8

      Normal Distribution

    9. Topic 9

      Combinations of Normal Distributions & Sample Mean Distributions

    10. Topic 10

      Poisson Distribution

    11. Topic 11

      Hypothesis Testing using the Chi-squared Distribution

    12. Topic 12

      Hypothesis Testing for Population Parameters

    13. Topic 13

      Transition Matrices & Markov Chains

    1. Topic 1

      Differentiation

    2. Topic 2

      Further Differentiation

    3. Topic 3

      Integration

    4. Topic 4

      Further Integration

    5. Topic 5

      Kinematics

    6. Topic 6

      Differential Equations

    7. Topic 7

      Coupled & Second Order Differential Equations