- #1
woundedtiger4
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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.
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.