What is a sequence of random variable?

In summary, this person is trying to answer a question about sequences of random variables, but is having trouble understanding the concept. They state that a sequence of random variables is a set of sequences in which the n'th number is chosen from the n'th random variable. They define a notation for sequences, and show how to create a sequence of five random numbers using the notation. They then state that if I toss the coin N times I get a sequence of random numbers. Finally, they state that all they are doing is defining a notation.
  • #1
woundedtiger4
188
0
Hi all,
I am really confused about the random variables
Toss a coin three times, so the set of possible outcomes is

Ω={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Define the random variables

X = Total number of heads, Y = Total number of tails

In symbol,

X(HHH)=3
X(HTT)=X(HTH)=X(THH)=2
X(HTT)=X(THT)=X(TTH)=1
X(TTT)=0

Y(TTT)=3
Y(TTH)=Y(THT)=Y(HTT)=2
Y(THH)=Y(HTH)=Y(HHT)=1
Y(HHH)=0

The probability of head on each toss is 1/2 and the probability of each element in Ω is 1/8, then:

P{ω∈Ω; X(ω)=0}=P{TTT}=1/8

P{ω∈Ω; X(ω)=1}=P{HTT,THT,THH}=3/8

P{ω∈Ω; X(ω)=2}=P{HHT, HTH,THH}=3/8

P{ω∈Ω; X(ω)=3}=P{HHH}=1/8


P{ω∈Ω; Y(ω)=0}=P{HHH}=1/8

P{ω∈Ω; Y(ω)=1}=P{THH,HTH,HHT}=3/8

P{ω∈Ω; Y(ω)=2}=P{TTH,THT,HTT}=3/8

P{ω∈Ω; Y(ω)=3}=P{TTT}=1/8

I have taken this example from text, now my question is that what is a sequence of random variable? The text says that the sequence of random variable is: X_1,X_2,X_3,...X_n. So in the above example, can we say that there are two sequence of variables which are,
X(HHH)=3 is X_1
X(HTT)=X(HTH)=X(THH)=2 is X_2
X(HTT)=X(THT)=X(TTH)=1 is X_3
X(TTT)=0 is X_4

Y(TTT)=3 is Y_1
Y(TTH)=Y(THT)=Y(HTT)=2 is Y_2
Y(THH)=Y(HTH)=Y(HHT)=1 is Y_3
Y(HHH)=0 is Y_4

OR

X is just one variable but taking different values so in the following
X(HHH)=3
X(HTT)=X(HTH)=X(THH)=2
X(HTT)=X(THT)=X(TTH)=1
X(TTT)=0
there is no sequence

Similarly Y is just one variable but taking different values so in the following
Y(TTT)=3
Y(TTH)=Y(THT)=Y(HTT)=2
Y(THH)=Y(HTH)=Y(HHT)=1
Y(HHH)=0

there is no sequence

or X,Y together forms a sequence?

I will really appreciate if someone can help me.

Thanks in advance.
 
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  • #2
what is a sequence of random variable
It is a sequence of numbers that come from an idealized random process - this means it is an abstract concept. How you tell is a particular sequence of numbers is random is a hard problem.

But I think I see your problem ...

Say I repeat the triple-coin-toss 5 times ... I get a sequence of 5 random numbers ... something like this perhaps: [itex]\{X_1, X_2, X_3, X_4, X_5\} = \{3, 0, 1, 1, 2\}[/itex] ... this is to say that [itex]X_n[/itex] is the result of the nth coin toss. If I did the experiment 20 times, I'd have [itex]X_1, X_2, X_3 \cdots[/itex] all the way to [itex]X_{20}[/itex] each one capable of having one of four distrete values. We can write [itex]X_n = x_n \in \{0,1,2,3\}[/itex] because, strictly speaking, each "X" (cap X) is a symbol that represents the act of tossing three coins and counting up the heads. The number of heads is usually represented by a lower-case "x".

That help?
 
Last edited:
  • #3
Simon Bridge said:
It is a sequence of numbers that come from an idealized random process - this means it is an abstract concept. How you tell is a particular sequence of numbers is random is a hard problem.

But I think I see your problem ...

Say I repeat the triple-coin-toss 5 times ... I get a sequence of 5 random numbers ... something like this perhaps: [itex]\{X_1, X_2, X_3, X_4, X_5\} = \{3, 0, 1, 1, 2\}[/itex] ... this is to say that [itex]X_n[/itex] is the result of the nth coin toss. If I did the experiment 20 times, I'd have [itex]X_1, X_2, X_3 \cdots[/itex] all the way to [itex]X_{20}[/itex] each one capable of having one of four distrete values. We can write [itex]X_n = x_n \in \{0,1,2,3\}[/itex] because, strictly speaking, each "X" (cap X) is a symbol that represents the act of tossing three coins and counting up the heads. The number of heads is usually represented by a lower-case "x".

That help?

Sorry, I didn't get it :(((((
 
  • #4
A sequence of random variables is just a set of sequences in which the n'th number is chosen from the n'th random variable. The sequence itself is a new random variable.
 
  • #5
Sorry, I didn't get it :(((((
Then I did not understand the question ... try restating it.
Let me give you a language to do that with:Let ##Z## be the event that I toss one coin (for simplicity) and count the number of "heads" that result.

Then the result of that toss can be ##z=1## for "heads" or ##z=0## for "not heads". So I can say that ##z \in \{0,1\}##.

If I toss the coin more than once, the ##Z_1## will be the first time I do it, ##Z_2## the second time, and so on.
Then the first result will be ##z_1## and the second result will be ##z_2## and so on.

If I toss the coin N times I get a sequence of random numbers.
Each member in the sequence can be a 1 or a 0.
The entire sequence will be the set of numbers:
##\{z_1, z_2, z_3, \cdots , z_{N-1}, z_N\}##

All I am doing here is defining a notation.

In this notation:
Z is a random process - a random number generator (if you will).
z is a random variable.
... some texts get a bit casual about the distinction.

Do you follow this so far?
 
Last edited:
  • #6
Simon Bridge said:
Then I did not understand the question ... try restating it.
Let me give you a language to do that with:


Let ##Z## be the event that I toss one coin (for simplicity) and count the number of "heads" that result.

Then the result of that toss can be ##z=1## for "heads" or ##z=0## for "not heads". So I can say that ##z \in \{0,1\}##.

If I toss the coin more than once, the ##Z_1## will be the first time I do it, ##Z_2## the second time, and so on.
Then the first result will be ##z_1## and the second result will be ##z_2## and so on.

If I toss the coin N times I get a sequence of random numbers.
Each member in the sequence can be a 1 or a 0.
The entire sequence will be the set of numbers:
##\{z_1, z_2, z_3, \cdots , z_{N-1}, z_N\}##

All I am doing here is defining a notation.

In this notation:
Z is a random process - a random number generator (if you will).
z is a random variable.
... some texts get a bit casual about the distinction.

Do you follow this so far?

yes now I do

Thank you sir
 
  • #7
Um ... OK. No worries then.
 

Related to What is a sequence of random variable?

1. What is a random variable sequence?

A random variable sequence is a sequence of random variables that are defined on the same probability space. These variables are usually denoted as X1, X2, X3, ... and can take on different values depending on the outcome of a random process.

2. What is the difference between a random variable and a random variable sequence?

A random variable is a single variable that represents the outcome of a random process. A random variable sequence, on the other hand, is a collection of random variables that are defined on the same probability space and are often used to model a series of random events or experiments.

3. How are random variable sequences used in statistics?

Random variable sequences are commonly used in statistics to analyze and model random phenomena. They can be used to calculate probabilities, estimate parameters, and make predictions about future outcomes.

4. Can a random variable sequence be infinite?

Yes, a random variable sequence can be infinite. This means that the sequence has an infinite number of random variables, each representing a different outcome of a random process. In some cases, an infinite random variable sequence may be used to model a continuous random variable.

5. What is the importance of understanding random variable sequences in research?

Random variable sequences are important in research because they allow scientists to model and analyze complex random phenomena in a systematic way. By understanding the properties and behavior of these sequences, researchers can make more accurate predictions and draw meaningful conclusions from their data.

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