Difference between revisions of "Science:MATH105 Probability/Lesson 1 DRV/1.01 Discrete Random Variables"

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''Note to MATH 105 instructors: I want to have a simple problem here that introduces the idea of probabilities as an area, and is appropriate for MATH 105 students. I'm not sure this one is a good problem, but I'd be happy to use another problem or modify this one to make it more applicable to MATH 105 students. Suggestions welcome! ''
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In many areas of science we are interested in quantifying  the '''probability''' that a certain outcome of an  experiment occurs. We can use a '''random variable''' to  identify numerical events that are of interest in an experiment. In this way, a random variable is a theoretical  representation of the physical or experimental process we  wish to study. More precisely, a random variable is a quantity without a fixed value, but which can assume  different values depending on how likely these values are to be observed; these likelihoods are probabilities.
  
==Discrete vs Continuous Random Variables ==
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To quantify the probability that a particular value, or  '''event''', occurs, we use a number between 0 and 1. A  probability of 0 implies that the event ''cannot'' occur,  whereas a probability of 1 implies that the event ''must''  occur.  Any value in the interval (0, 1) means that the  event will only occur some of the time. Equivalently, if an  event occurs with probability ''p'', then this means there  is a ''p''(100)% chance of observing this event.
  
''to go on the "About" page", or elsewhere''
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Conventionally, we denote random variables by capital  letters, and particular values that they can assume by  lowercase letters. So we can say that ''X'' is a random  variable that can assume certain particular values  ''x'' with certain probabilities. 
  
In MATH 105, we deal with
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We use the notation Pr(''X'' = ''x'') to denote the  probability that the random variable ''X'' assumes the  particular value ''x''. The range of ''x'' for which this  expression makes sense is of course dependent on the  possible values of the random variable ''X''. We distinguish  between two key cases.
  
* discrete random variables, and
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If ''X'' can assume only finitely many or countably many  values, then we say that ''X'' is a '''discrete random variable'''. Saying that ''X'' can assume only ''finitely  many or countably many'' values means that we should be able  to ''list'' the possible values for the random variable  ''X''. If this list is finite, we can say that ''X'' may  take any value from the list ''x<sub>1</sub>'',  ''x<sub>2</sub>'',..., ''x<sub>n</sub>'', for some positive  integer ''n''. If the list is (countably) infinite, we can  list the possible values for ''X'' as ''x<sub>1</sub>'',  ''x<sub>2</sub>'',.... This is then a list without end (for  example, the list of all positive integers).
* continuous random variables.  
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You will notice that certain definitions and concepts between them will be related.  
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{| border="1" cellspacing="0" cellpadding="4" align="center"
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|- style="background-color:#f0f0f0;"
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! Discrete Random Variables
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|-
 +
|
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# A discrete random variable ''X'' is a quantity that can  assume any value ''x'' from a discrete list of values with a  certain probability.
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# The probability that the random variable ''X'' assumes the  particular value ''x'' is denoted by Pr(''X'' = ''x''). This  collection of probabilities, along with all possible values  ''x'', is the '''probability distribution''' of the random  variable ''X''.
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# A discrete list of values is any collection of values that  is finite or countably infinite (i.e. can be written in a  list).
 +
|}
  
These concepts are introduced assuming no prior knowledge of probability, but they are discussed with an assumption of integration. That is, probability is introduced as an application of of integration.  
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This terminology is in contrast to a '''continuous random  variable''', where the values the random variable can assume  are given by a continuum of values. For example, we could  define a random variable that can take any value in the  interval [1,2]. The values ''X'' can assume are then any  real number in [1,2]. We will discuss continuous random  variables in detail in the second part of this module. For  now, we deal strictly with discrete random variables.
  
==Discrete Random Variables==
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We state a few facts that should be intuitively obvious for  probabilities in general. Namely, the chance of some  particular event occurring should always be nonnegative and  no greater than 100%. Also, the chance that ''something''  happens should be certain. From these facts, we can conclude  that the chance of witnessing a particular event should be  100% less the chance of seeing ''anything but'' that  particular event.
  
In any experiment, only one '''outcome''' occurs. If we flip a coin, the outcome will be either "heads" (H) or "tails" (T). If the coin is a "fair" coin, it is equally likely that the coin will land as tails or heads.
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{| border="1" cellspacing="0" cellpadding="4" align="center"
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|- style="background-color:#f0f0f0;"
 +
! Discrete Probability Rules
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|-
 +
|
 +
# Probabilities are numbers between 0 and 1: 0 ≤ Pr(''X'' ''x<sub>k</sub>'') ≤ 1 for all ''k''
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# The sum of all probabilities for a given experiment  (random variable) is equal to one: <math>\sum_k \text{Pr}(X  = x_k) = 1\!</math>
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# The probability of an event is 1 minus the probability  that any other event occurs: <math>\text{Pr}(X = x_n) = 1 -  \sum_{k\neq n}\text{Pr}(X = x_k)</math>
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|}
  
  
We are often interested in quantifying the '''probability''' that a certain outcome occurs. Suppose we toss a fair coin tossed two times, and want to the probability that the coin will land as tails both times.
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==Example: Tossing a Fair Coin Once==
  
There are four possible outcomes:
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If we toss a coin into the air, there are only two possible  outcomes: it will land as either "heads" (H) or "tails" (T).  If the tossed coin is a "fair" coin, it is equally likely  that the coin will land as tails or as heads. In other  words, there is a 50% chance (1/2 probability) that the coin  will land heads, and a 50% chance (1/2 probability) that the  coin will land tails. Notice that the sum of these  probabilities is 1 and that each probability is a number in  the interval [0,1].
 +
 
 +
We can define the random variable ''X'' to represent this  coin tossing experiment.  That is, we define ''X'' to be the  discrete random variable that takes the value 0 with  probability 1/2 and takes the value 1 with probability 1/2.  Notice that with this notation, the experimental event that  "we toss a fair coin and observe heads" is the same as the  theoretical event that "the random variable ''X'' is  observed to take the value 0"; i.e. we identify the number 0  with the outcome of "heads", and identify the number 1 with  the outcome of "tails". We say that ''X'' is a '''Bernoulli  random variable''' with parameter 1/2 and can write ''X'' ~  Ber(1/2).
 +
 
 +
==Example: Tossing a Fair Coin Twice==
 +
 
 +
Similarly, if we toss a fair coin two times, there are four possible outcomes. Each outcome is a sequence of heads (H)  or tails (T):
 +
 
 +
* HH
 +
* HT
 +
* TH
 +
* TT
 +
 
 +
Because the coin is fair, each outcome is equally likely to  occur. There are 4 possible outcomes, so we assign each  outcome a probability of 1/4.
 +
 
 +
Equivalently, we notice that for any of the four possible  events to occur, we must observe two distinct events from  two separate flips of a fair coin.  So for example, to  observe the sequence HH, we must flip a fair coin once and  observe H, then flip a fair coin again and observe H once  again. (We say that these two events are '''independent'''  since the outcome of one event has no effect on the outcome  of the other.) Since the probability of observing H after a  flip of a fair coin is 1/2, we see that the probability of  observing the sequence HH should be (1/2)×(1/2) = 1/4.
 +
 
 +
Observe that again, all of our probabilities sum to 1, and  each probability is a number on the interval [0, 1]. Just as  before, we can identify each outcome of our experiment with a numerical value. Let us make the following assignments:
 +
 
 +
* HH -> 0
 +
* HT -> 1
 +
* TH -> 2
 +
* TT -> 3
 +
 
 +
This assignment defines a numerical discrete random variable ''Y'' that represents our coin tossing experiment. We see that ''Y'' takes the  value 0 with probability 1/4,  1 with probability 1/4, 2  with probability 1/4, and 3 with probability 1/4. Using our  general notation to describe this probability distribution,  we can summarize by writing
 +
 
 +
<math> \text{Pr}(Y = k) = 1/4,\text{ for } k = 1,2,3,4.  </math>
 +
 
 +
Notice that with this notation, the experimental event that  "we toss two fair coins and observe first tails, then heads"  is the same as the theoretical event that "the random  variable ''Y'' is observed to take the value 2". We say that  ''Y'' is a '''uniform discrete random variable''' with  parameter 4 since ''Y'' takes each of its four possible  values with equal, or uniform, probability. To denote this  distributional relationship, we can write ''Y'' ~  Uniform(4).
  
HH
 
HT
 
TH
 
TT
 
  
Because the coin is fair, each outcome is equally likely to occur.
 
  
  
 
 
[[Category:MATH105]]
 
[[Category:MATH105]]
 
[[Category:MATH105 Probability]]
 
[[Category:MATH105 Probability]]
 
[[Category:MATH105 Lesson 1]]
 
[[Category:MATH105 Lesson 1]]

Latest revision as of 17:01, 15 February 2012

In many areas of science we are interested in quantifying the probability that a certain outcome of an experiment occurs. We can use a random variable to identify numerical events that are of interest in an experiment. In this way, a random variable is a theoretical representation of the physical or experimental process we wish to study. More precisely, a random variable is a quantity without a fixed value, but which can assume different values depending on how likely these values are to be observed; these likelihoods are probabilities.

To quantify the probability that a particular value, or event, occurs, we use a number between 0 and 1. A probability of 0 implies that the event cannot occur, whereas a probability of 1 implies that the event must occur. Any value in the interval (0, 1) means that the event will only occur some of the time. Equivalently, if an event occurs with probability p, then this means there is a p(100)% chance of observing this event.

Conventionally, we denote random variables by capital letters, and particular values that they can assume by lowercase letters. So we can say that X is a random variable that can assume certain particular values x with certain probabilities.

We use the notation Pr(X = x) to denote the probability that the random variable X assumes the particular value x. The range of x for which this expression makes sense is of course dependent on the possible values of the random variable X. We distinguish between two key cases.

If X can assume only finitely many or countably many values, then we say that X is a discrete random variable. Saying that X can assume only finitely many or countably many values means that we should be able to list the possible values for the random variable X. If this list is finite, we can say that X may take any value from the list x1, x2,..., xn, for some positive integer n. If the list is (countably) infinite, we can list the possible values for X as x1, x2,.... This is then a list without end (for example, the list of all positive integers).

Discrete Random Variables
  1. A discrete random variable X is a quantity that can assume any value x from a discrete list of values with a certain probability.
  2. The probability that the random variable X assumes the particular value x is denoted by Pr(X = x). This collection of probabilities, along with all possible values x, is the probability distribution of the random variable X.
  3. A discrete list of values is any collection of values that is finite or countably infinite (i.e. can be written in a list).

This terminology is in contrast to a continuous random variable, where the values the random variable can assume are given by a continuum of values. For example, we could define a random variable that can take any value in the interval [1,2]. The values X can assume are then any real number in [1,2]. We will discuss continuous random variables in detail in the second part of this module. For now, we deal strictly with discrete random variables.

We state a few facts that should be intuitively obvious for probabilities in general. Namely, the chance of some particular event occurring should always be nonnegative and no greater than 100%. Also, the chance that something happens should be certain. From these facts, we can conclude that the chance of witnessing a particular event should be 100% less the chance of seeing anything but that particular event.

Discrete Probability Rules
  1. Probabilities are numbers between 0 and 1: 0 ≤ Pr(X = xk) ≤ 1 for all k
  2. The sum of all probabilities for a given experiment (random variable) is equal to one:
  3. The probability of an event is 1 minus the probability that any other event occurs:


Example: Tossing a Fair Coin Once

If we toss a coin into the air, there are only two possible outcomes: it will land as either "heads" (H) or "tails" (T). If the tossed coin is a "fair" coin, it is equally likely that the coin will land as tails or as heads. In other words, there is a 50% chance (1/2 probability) that the coin will land heads, and a 50% chance (1/2 probability) that the coin will land tails. Notice that the sum of these probabilities is 1 and that each probability is a number in the interval [0,1].

We can define the random variable X to represent this coin tossing experiment. That is, we define X to be the discrete random variable that takes the value 0 with probability 1/2 and takes the value 1 with probability 1/2. Notice that with this notation, the experimental event that "we toss a fair coin and observe heads" is the same as the theoretical event that "the random variable X is observed to take the value 0"; i.e. we identify the number 0 with the outcome of "heads", and identify the number 1 with the outcome of "tails". We say that X is a Bernoulli random variable with parameter 1/2 and can write X ~ Ber(1/2).

Example: Tossing a Fair Coin Twice

Similarly, if we toss a fair coin two times, there are four possible outcomes. Each outcome is a sequence of heads (H) or tails (T):

  • HH
  • HT
  • TH
  • TT

Because the coin is fair, each outcome is equally likely to occur. There are 4 possible outcomes, so we assign each outcome a probability of 1/4.

Equivalently, we notice that for any of the four possible events to occur, we must observe two distinct events from two separate flips of a fair coin. So for example, to observe the sequence HH, we must flip a fair coin once and observe H, then flip a fair coin again and observe H once again. (We say that these two events are independent since the outcome of one event has no effect on the outcome of the other.) Since the probability of observing H after a flip of a fair coin is 1/2, we see that the probability of observing the sequence HH should be (1/2)×(1/2) = 1/4.

Observe that again, all of our probabilities sum to 1, and each probability is a number on the interval [0, 1]. Just as before, we can identify each outcome of our experiment with a numerical value. Let us make the following assignments:

  • HH -> 0
  • HT -> 1
  • TH -> 2
  • TT -> 3

This assignment defines a numerical discrete random variable Y that represents our coin tossing experiment. We see that Y takes the value 0 with probability 1/4, 1 with probability 1/4, 2 with probability 1/4, and 3 with probability 1/4. Using our general notation to describe this probability distribution, we can summarize by writing

Notice that with this notation, the experimental event that "we toss two fair coins and observe first tails, then heads" is the same as the theoretical event that "the random variable Y is observed to take the value 2". We say that Y is a uniform discrete random variable with parameter 4 since Y takes each of its four possible values with equal, or uniform, probability. To denote this distributional relationship, we can write Y ~ Uniform(4).