AP Stats: Chapter 6.1
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
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 | x⌄1 | x⌄2 | x⌄3 | … |
---|---|---|---|---|
probability | p⌄1 | p⌄2 | p⌄3 | … |
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
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.
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 | x⌄1 | x⌄2 | x⌄3 | … |
---|---|---|---|---|
probability | p⌄1 | p⌄2 | p⌄3 | … |
To find the mean (expected value) of X, multiply each possible value of X by its probability then add all of the products.
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^2⌄x = (X⌄1 - μ⌄x)^2* p⌄1 + (X⌄2 - μ⌄x)^2* p⌄2 + (X⌄3 - 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.
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
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)= ∑ x⌄i * p⌄i = x⌄1 p⌄1 + x⌄2 * p⌄2 + x⌄3 * p⌄3 + ···
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=∑ (x⌄i−μ⌄X)^2*p⌄i = (x⌄1−μ⌄X)^2 * p⌄1 + (x⌄2−μ⌄X)^2 p⌄2 + (xV3−μ⌄X)^2 * p⌄3 + ...
The standard deviation of X is the square root of the variance.
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
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 | x⌄1 | x⌄2 | x⌄3 | … |
---|---|---|---|---|
probability | p⌄1 | p⌄2 | p⌄3 | … |
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
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.
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 | x⌄1 | x⌄2 | x⌄3 | … |
---|---|---|---|---|
probability | p⌄1 | p⌄2 | p⌄3 | … |
To find the mean (expected value) of X, multiply each possible value of X by its probability then add all of the products.
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^2⌄x = (X⌄1 - μ⌄x)^2* p⌄1 + (X⌄2 - μ⌄x)^2* p⌄2 + (X⌄3 - 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.
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
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)= ∑ x⌄i * p⌄i = x⌄1 p⌄1 + x⌄2 * p⌄2 + x⌄3 * p⌄3 + ···
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=∑ (x⌄i−μ⌄X)^2*p⌄i = (x⌄1−μ⌄X)^2 * p⌄1 + (x⌄2−μ⌄X)^2 p⌄2 + (xV3−μ⌄X)^2 * p⌄3 + ...
The standard deviation of X is the square root of the variance.