ผลต่างระหว่างรุ่นของ "Probstat/notes/random variables"
Jittat (คุย | มีส่วนร่วม) |
Jittat (คุย | มีส่วนร่วม) |
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| แถว 1: | แถว 1: | ||
: ''This is part of [[probstat]].'' | : ''This is part of [[probstat]].'' | ||
| − | In many cases, after we perform a random experiment, we are interested in certain quantity from the outcome, not the actual outcome. In that case, we can define a ''random variable'', which is a function from the sample space to real numbers, to represent the random quantity that we are interested in. | + | In many cases, after we perform a random experiment, we are interested in certain quantity from the outcome, not the actual outcome. In that case, we can define a '''random variable''', which is a function from the sample space to real numbers, to represent the random quantity that we are interested in. |
For example, consider the following experiment. We toss two dice. Let a random variable ''X'' be the sum of the values of these two dice. The table below shows the outcomes and probabilities related to ''X''. | For example, consider the following experiment. We toss two dice. Let a random variable ''X'' be the sum of the values of these two dice. The table below shows the outcomes and probabilities related to ''X''. | ||
| แถว 59: | แถว 59: | ||
Therefore, it is reasonable to consider the probability of events defined by random variables. From the two-dice example, we have ''P''{ ''X >= 11'' } = ''P''({(5,6), (6,5), (6,6)}) = 3/36. | Therefore, it is reasonable to consider the probability of events defined by random variables. From the two-dice example, we have ''P''{ ''X >= 11'' } = ''P''({(5,6), (6,5), (6,6)}) = 3/36. | ||
| − | Given a random variable ''X'', a probability mass function ''p'' of ''X'' is defined as ''p(i)'' = ''P''{ ''X'' = ''i'' }. We usually denote the probability mass function as ''pmf''. | + | Given a random variable ''X'', a '''probability mass function''' ''p'' of ''X'' is defined as ''p(i)'' = ''P''{ ''X'' = ''i'' }. We usually denote the probability mass function as ''pmf''. |
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| + | === Expectation === | ||
รุ่นแก้ไขเมื่อ 02:39, 18 กันยายน 2557
- This is part of probstat.
In many cases, after we perform a random experiment, we are interested in certain quantity from the outcome, not the actual outcome. In that case, we can define a random variable, which is a function from the sample space to real numbers, to represent the random quantity that we are interested in.
For example, consider the following experiment. We toss two dice. Let a random variable X be the sum of the values of these two dice. The table below shows the outcomes and probabilities related to X.
| i | Outcomes for which X = i | Probability P{ X = i } |
| 2 | (1,1) | 1/36 |
| 3 | (1,2), (2,1) | 2/36 |
| 4 | (1,3), (2,2), (3,1) | 3/36 |
| 5 | (1,4), (2,3), (3,2), (4,1) | 4/36 |
| 6 | (1,5), (2,4), (3,3), (4,2), (5,1) | 5/36 |
| 7 | (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) | 6/36 |
| 8 | (2,6), (3,5), (4,4), (5,3), (6,2) | 5/36 |
| 9 | (3,6), (4,5), (5,4), (6,4) | 4/36 |
| 10 | (4,6), (5,5), (6,4) | 3/36 |
| 11 | (5,6), (6,5) | 2/36 |
| 12 | (6,6) | 1/36 |
A random variable X also induces events related to it. From the previous example, the event that X=10 corresponds to the subset {(4,6), (5,5), 6,4)} of the sample space. Also, if the event X >= 11 corresponds to {(5,6), (6,5), (6,6)}.
Therefore, it is reasonable to consider the probability of events defined by random variables. From the two-dice example, we have P{ X >= 11 } = P({(5,6), (6,5), (6,6)}) = 3/36.
Given a random variable X, a probability mass function p of X is defined as p(i) = P{ X = i }. We usually denote the probability mass function as pmf.
