AP Statistics Curriculum 2007 Distrib RV

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 Revision as of 18:45, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 02:33, 30 January 2008 (view source)IvoDinov (Talk | contribs) Newer edit → Line 1: Line 1: ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Random Variables and Probability Distributions== ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Random Variables and Probability Distributions== - === Random Variables and Probability Distributions=== + === Random Variables=== - Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)! + A '''random variable''' is a function or a mapping from a sample space into the real numbers (most of the time). In other words, a random variable assignes real values to outcomes of experiments. This mapping is called ''random'', as the output values of the mapping depend on the outcome of the experiment, which are indeed random. So, instead of studying the raw outcomes of experiments (e.g., define and compute probabilities), most of the time we study (or compute probabilities) on the corresponding random variables instead. The [http://en.wikipedia.org/wiki/Random_variable formal general definition of random variables may be found here]. -
[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]
+ - ===Approach=== + ===Examples of random variables=== - Models & strategies for solving the problem, data understanding & inference. + - * TBD + * '''Die''': In rolling a regular hexagonal die, the sample space is clearly and numerically well-defined and in this case the random variable is the identity function assigning to each face of the die the numerical value it represents. This the possible outcomes of the RV of this experiment are { 1, 2, 3, 4, 5, 6 }. You can see this explicit RV mapping in the [[SOCR_EduMaterials_Activities_DiceExperiment | SOCR Die Experiment]]. + + * '''Coin''': For a coin toss, a suitable space of possible outcomes is S={H, T} (for heads and tails).  In this case these are not numerical values, so we can define a RV that maps these to numbers. For instance, we can define the RV $X: S \longrightarrow [0, 1]$ as: $X(s) = \begin{cases}0,& s = \texttt{H},\\ + 1,& s = \texttt{T}.\end{cases}$. You can see this explicit RV mapping of heads and tails to numbers in the [[SOCR_EduMaterials_Activities_BinomialCoinExperiment | SOCR Coin Experiment]]. + + * '''Card''': Suppose we draw a [[ | 5-card hand from a standard 52-card deck]] and we are interested in the probability that the hand contains at least one pair of cards with identical denomination. Then the sample space of this experimetn is large - it sould be difficult to list all possible outcomes. However, we can assign a random variable $X(s) = \begin{cases}0,& s = \texttt{no-pair},\\ + 1,& s = \texttt{at least one pair}.\end{cases}$ and try to compute the probability of P(X=1), i., the chance that the hand contains a pair. You can see this explicit RV mapping and the calculations of this probability at the [[SOCR_EduMaterials_Activities_CardExperiment | SOCR Card Experiment]]. + + + ===How do we use RVs?=== + There are 3 important quantities that we are always interested in when we study random processes. each of these may be phrased in terms of RVs, which simplifies their calculations. + + * '''Probability distribution''': What is the probability of $P(X=x_o)$? For instance, in the card example above, we may be interested in [[SOCR_EduMaterials_Activities_CardExperiment#Applications | P(at least 1 pair) = P(X=1) = P(1 pair only) = 0.422569]]. Or in the die example, we may want to know P(Even number turns up) = $P(X \in \{2, 4, 6 \}) = 0.5$. + + * '''Cummulative distribution''': [itex]P(X + {| class="wikitable" style="text-align:center; width:75%" border="1" + |- + | x || 0 || 1 || 2 || 3 || 4 || 5 || 6 + |- + | (P(X<=x) || 1/6 || 2/6 || 3/6 || 4/6 || 5/6 || 1 + |} + + + * M'''oments/expected values''': ===Model Validation=== ===Model Validation===

General Advance-Placement (AP) Statistics Curriculum - Random Variables and Probability Distributions

Random Variables

A random variable is a function or a mapping from a sample space into the real numbers (most of the time). In other words, a random variable assignes real values to outcomes of experiments. This mapping is called random, as the output values of the mapping depend on the outcome of the experiment, which are indeed random. So, instead of studying the raw outcomes of experiments (e.g., define and compute probabilities), most of the time we study (or compute probabilities) on the corresponding random variables instead. The formal general definition of random variables may be found here.

Examples of random variables

• Die: In rolling a regular hexagonal die, the sample space is clearly and numerically well-defined and in this case the random variable is the identity function assigning to each face of the die the numerical value it represents. This the possible outcomes of the RV of this experiment are { 1, 2, 3, 4, 5, 6 }. You can see this explicit RV mapping in the SOCR Die Experiment.
• Coin: For a coin toss, a suitable space of possible outcomes is S={H, T} (for heads and tails). In this case these are not numerical values, so we can define a RV that maps these to numbers. For instance, we can define the RV $X: S \longrightarrow [0, 1]$ as: $X(s) = \begin{cases}0,& s = \texttt{H},\\ 1,& s = \texttt{T}.\end{cases}$. You can see this explicit RV mapping of heads and tails to numbers in the SOCR Coin Experiment.
• Card: Suppose we draw a [[ | 5-card hand from a standard 52-card deck]] and we are interested in the probability that the hand contains at least one pair of cards with identical denomination. Then the sample space of this experimetn is large - it sould be difficult to list all possible outcomes. However, we can assign a random variable $X(s) = \begin{cases}0,& s = \texttt{no-pair},\\ 1,& s = \texttt{at least one pair}.\end{cases}$ and try to compute the probability of P(X=1), i., the chance that the hand contains a pair. You can see this explicit RV mapping and the calculations of this probability at the SOCR Card Experiment.

How do we use RVs?

There are 3 important quantities that we are always interested in when we study random processes. each of these may be phrased in terms of RVs, which simplifies their calculations.

• Probability distribution: What is the probability of P(X = xo)? For instance, in the card example above, we may be interested in P(at least 1 pair) = P(X=1) = P(1 pair only) = 0.422569. Or in the die example, we may want to know P(Even number turns up) = $P(X \in \{2, 4, 6 \}) = 0.5$.
• Cummulative distribution: P(X < xo), for all xo. For instance, for the die example we have the following discrete cummulative distribution table:
 x 0 1 2 3 4 5 6 (P(X<=x) 1/6 2/6 3/6 4/6 5/6 1
• Moments/expected values:

Model Validation

Checking/affirming underlying assumptions.

• TBD

• TBD

Examples

Computer simulations and real observed data.

• TBD

Hands-on activities

Step-by-step practice problems.

• TBD

• TBD