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AP Stats: Chapter 6.1

Introduction


  • A random variable takes numerical values that describe the outcomes of a chance process

    • Use capital letters to designate random variables

  • The probability distribution of a random variable gives it possible values and their probabilities

    • Two main types: discrete and continuous

Discrete Random Variables


  • A discrete random variable, X, takes a fixed set of possible values with gaps between them

  • Probability Distribution for a Discrete Random Variable

    • The probability distribution of a discrete random variable, X, list the values x, and their probabilities p:

value

x1

x2

x3

probability

p1

p2

p3

  • For the probability distribution to be valid, the probabilities must satisfy two requirements:

    • every probability, p, is a number between 1 and 0

    • the sum of the probabilities is 1

Analyzing Discrete Random Variables: Describing Change


  • Display probability distribution with a histogram

    • Values of variable go on the horizontal axis

    • Probability goes on the vertical axis

  • A probability distribution histogram is really just a relative frequency histogram because probabilities are long-run frequencies.

Measuring Center: The Mean (Expected Value) of a Discrete Random Variable


  • The mean x̄ of a quantitative data set with n observations is

    • x̄ = sum of data values/number of data values

  • The mean of any discrete random variable is an average of the possible outcomes, but a weighted average in which each outcome is weighted by its probability

  • The mean (expected value) of a discrete random variable is its average value over many, many repetitions of the same chance process.

    • Suppose X is a discrete random variable with probability distribution:

value

x1

x2

x3

probability

p1

p2

p3

  • To find the mean (expected value) of X, multiply each possible value of X by its probability then add all of the products.

Measuring Variability: The Standard Deviation (and Variance) of a Discrete Random Variable


  • The standard deviation of a discrete random variable measures how much the values of the variable typically vary from the mean.

    • Suppose that X is a discrete random variable with the probability distribution

    • and that μ⌄x is the mean of X. The variance of X is

      • Var(X)=σ^2x = (X1 - μ⌄x)^2* p⌄1 + (X⌄2 - μ⌄x)^2* p⌄2 + (X3 - μ⌄x)^2* p⌄3 + …

    • The standard of deviation of X is the square root of the variance.

  • Using a Calculator to Analyze Discrete Random Variables: TI-83/84

    • Enter the values of the random variable list in L1 and the corresponding probabilities in list L2

    • To graph a histogram of the probability distribution:

      • In the stat plot menu define Plot 1 to be a histogram with Xlist: L1 and Freq: L2

      • Adjust your window settings as follows: Xmin = -1, Xmax = 11, Xscl = 1, Ymin = -0.1, Ymax = 0.5, Yscl = 0.1.

      • Press GRAPH

    • To calculate the mean and standard deviation of the random variable, use one-variable statistics with the values in L1 and the probabilities (relative frequencies) in L2. Press STAT, arrow to the CALC menu, and choose 1-Var Stats.

      • OS 2.55 or later: In the dialog box, specify List: L1 and FreqList: L2. Then choose Calculate.

      • Older OS: Execute the command 1-Var Stats L1,L2.

Continuous Random Variables


  • A continuous random variable can take any value in an interval on the number line

  • How to Find Probabilities for a Continuous Random Variable

    • The probability of any event involving a continuous random variable is the area under the density curve and directly above the values on the horizontal axis that make up the event

  • The probability for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes

Summary


  • A random variable takes numerical values determined by the outcome of a chance process. The probability distribution of a random variable gives its possible values and their probabilities. There are two types of random variables: discrete and continuous.

  • A discrete random variable has a fixed set of possible values with gaps between them.

    • A valid probability distribution assigns each of these values a probability between 0 and 1 such that the sum of all the probabilities is exactly 1.

    • The probability of any event is the sum of the probabilities of all the values that make up the event.

    • We can display the probability distribution as a histogram, with the values of the variable on the horizontal axis and the probabilities on the vertical axis.

  • A continuous random variable can take any value in an interval on the number line.

    • A valid probability distribution for a continuous random variable is described by a density curve with area 1 underneath.

    • The probability of any event is the area under the density curve directly above the values on the horizontal axis that make up the event.

  • We can describe the shape of a probability distribution histogram or density curve in the same way as we did a distribution of quantitative data—by identifying symmetry or skewness and any major peaks.

  • Use the mean to summarize the center of a probability distribution. The mean of a random variable μX is the balance point of the probability distribution histogram or density curve.

    • The mean is the long-run average value of the variable after many repetitions of the chance process. It is also known as the expected value of the random variable, E(X).

    • If X is a discrete random variable, the mean is the average of the values of X, each weighted by its probability:

      • μX=E(X)= ∑ xi * pi = x1 p1 + x2 * p2 + x3 * p3 + ···

  • Use the standard deviation to summarize the variability of a probability distribution. The standard deviation of a random variable σX measures how much the values of the variable typically vary from the mean.

    • If X is a discrete random variable, the variance of X is the “average” squared deviation of the values of the variable from their mean:

      • σX2=∑ (xi−μX)^2*pi = (x1−μX)^2 * p1 + (x2−μX)^2 p2 + (xV3−μX)^2 * p3 + ...

  • The standard deviation of X is the square root of the variance.

AP Stats: Chapter 6.1

Introduction


  • A random variable takes numerical values that describe the outcomes of a chance process

    • Use capital letters to designate random variables

  • The probability distribution of a random variable gives it possible values and their probabilities

    • Two main types: discrete and continuous

Discrete Random Variables


  • A discrete random variable, X, takes a fixed set of possible values with gaps between them

  • Probability Distribution for a Discrete Random Variable

    • The probability distribution of a discrete random variable, X, list the values x, and their probabilities p:

value

x1

x2

x3

probability

p1

p2

p3

  • For the probability distribution to be valid, the probabilities must satisfy two requirements:

    • every probability, p, is a number between 1 and 0

    • the sum of the probabilities is 1

Analyzing Discrete Random Variables: Describing Change


  • Display probability distribution with a histogram

    • Values of variable go on the horizontal axis

    • Probability goes on the vertical axis

  • A probability distribution histogram is really just a relative frequency histogram because probabilities are long-run frequencies.

Measuring Center: The Mean (Expected Value) of a Discrete Random Variable


  • The mean x̄ of a quantitative data set with n observations is

    • x̄ = sum of data values/number of data values

  • The mean of any discrete random variable is an average of the possible outcomes, but a weighted average in which each outcome is weighted by its probability

  • The mean (expected value) of a discrete random variable is its average value over many, many repetitions of the same chance process.

    • Suppose X is a discrete random variable with probability distribution:

value

x1

x2

x3

probability

p1

p2

p3

  • To find the mean (expected value) of X, multiply each possible value of X by its probability then add all of the products.

Measuring Variability: The Standard Deviation (and Variance) of a Discrete Random Variable


  • The standard deviation of a discrete random variable measures how much the values of the variable typically vary from the mean.

    • Suppose that X is a discrete random variable with the probability distribution

    • and that μ⌄x is the mean of X. The variance of X is

      • Var(X)=σ^2x = (X1 - μ⌄x)^2* p⌄1 + (X⌄2 - μ⌄x)^2* p⌄2 + (X3 - μ⌄x)^2* p⌄3 + …

    • The standard of deviation of X is the square root of the variance.

  • Using a Calculator to Analyze Discrete Random Variables: TI-83/84

    • Enter the values of the random variable list in L1 and the corresponding probabilities in list L2

    • To graph a histogram of the probability distribution:

      • In the stat plot menu define Plot 1 to be a histogram with Xlist: L1 and Freq: L2

      • Adjust your window settings as follows: Xmin = -1, Xmax = 11, Xscl = 1, Ymin = -0.1, Ymax = 0.5, Yscl = 0.1.

      • Press GRAPH

    • To calculate the mean and standard deviation of the random variable, use one-variable statistics with the values in L1 and the probabilities (relative frequencies) in L2. Press STAT, arrow to the CALC menu, and choose 1-Var Stats.

      • OS 2.55 or later: In the dialog box, specify List: L1 and FreqList: L2. Then choose Calculate.

      • Older OS: Execute the command 1-Var Stats L1,L2.

Continuous Random Variables


  • A continuous random variable can take any value in an interval on the number line

  • How to Find Probabilities for a Continuous Random Variable

    • The probability of any event involving a continuous random variable is the area under the density curve and directly above the values on the horizontal axis that make up the event

  • The probability for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes

Summary


  • A random variable takes numerical values determined by the outcome of a chance process. The probability distribution of a random variable gives its possible values and their probabilities. There are two types of random variables: discrete and continuous.

  • A discrete random variable has a fixed set of possible values with gaps between them.

    • A valid probability distribution assigns each of these values a probability between 0 and 1 such that the sum of all the probabilities is exactly 1.

    • The probability of any event is the sum of the probabilities of all the values that make up the event.

    • We can display the probability distribution as a histogram, with the values of the variable on the horizontal axis and the probabilities on the vertical axis.

  • A continuous random variable can take any value in an interval on the number line.

    • A valid probability distribution for a continuous random variable is described by a density curve with area 1 underneath.

    • The probability of any event is the area under the density curve directly above the values on the horizontal axis that make up the event.

  • We can describe the shape of a probability distribution histogram or density curve in the same way as we did a distribution of quantitative data—by identifying symmetry or skewness and any major peaks.

  • Use the mean to summarize the center of a probability distribution. The mean of a random variable μX is the balance point of the probability distribution histogram or density curve.

    • The mean is the long-run average value of the variable after many repetitions of the chance process. It is also known as the expected value of the random variable, E(X).

    • If X is a discrete random variable, the mean is the average of the values of X, each weighted by its probability:

      • μX=E(X)= ∑ xi * pi = x1 p1 + x2 * p2 + x3 * p3 + ···

  • Use the standard deviation to summarize the variability of a probability distribution. The standard deviation of a random variable σX measures how much the values of the variable typically vary from the mean.

    • If X is a discrete random variable, the variance of X is the “average” squared deviation of the values of the variable from their mean:

      • σX2=∑ (xi−μX)^2*pi = (x1−μX)^2 * p1 + (x2−μX)^2 p2 + (xV3−μX)^2 * p3 + ...

  • The standard deviation of X is the square root of the variance.

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