ผลต่างระหว่างรุ่นของ "Probstat/notes/random variables"

จาก Theory Wiki
ไปยังการนำทาง ไปยังการค้นหา
แถว 58: แถว 58:
  
 
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''.

รุ่นแก้ไขเมื่อ 02:38, 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.