- #1

- 189

- 0

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

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.

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.