What is a sequence of random variable?

[woundedtiger4]

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.

 

[Simon Bridge]

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: $\{X_1, X_2, X_3, X_4, X_5\} = \{3, 0, 1, 1, 2\}$ ... this is to say that $X_n$ is the result of the nth coin toss. If I did the experiment 20 times, I'd have $X_1, X_2, X_3 \cdots$ all the way to $X_{20}$ each one capable of having one of four distrete values. We can write $X_n = x_n \in \{0,1,2,3\}$ 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?

 

[woundedtiger4]

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: $\{X_1, X_2, X_3, X_4, X_5\} = \{3, 0, 1, 1, 2\}$ ... this is to say that $X_n$ is the result of the nth coin toss. If I did the experiment 20 times, I'd have $X_1, X_2, X_3 \cdots$ all the way to $X_{20}$ each one capable of having one of four distrete values. We can write $X_n = x_n \in \{0,1,2,3\}$ 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 :(((((

 

[lavinia]

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.

 

[Simon Bridge]

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?

 

[woundedtiger4]

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

 

[Simon Bridge]

Um ... OK. No worries then.