Question a):
You flip a fair coin $7$ times; these coin flips are independent of each other.
Define the random variables:
$X$ = the number of heads in these $7$ coin flips, and
$Y$ = the number of tails in these $7$ coin flips.
Are the random variables independent or not? Show why or why not?
Attempt:
I know there are $2^{7}$ = $128$ possible sequences for the con toss $7$ times.
{HHHHHHT, HHHHHTT, HHHHTTT, HHHTTTT, HHTTTTT, HTTTTTT, HHHHHHH}
If I take $P(X=1 $ and $ Y=1) = P(X=1)P(Y=1)$
The Probability of just 1 head and 1 tail would be $\frac{7}{128}$ for both.
The Probability of just 1 head and 1 tail occurring together is $0$
So, they are not equal and hence dependent. Is this correct way?
Question b):
Consider the set $(1,2,3,…10)$. You choose a uniformly random element z in S.
Define the random variables:
$X$ = $0$ if z is even and $1$ is z odd
$Y$ = $0$ if z {1,2}, $1$ if z {3,4,5,6}, $2$ if z {7,8,9,20}
Are the random variables independent or not? Show why or why not?
Attempt:
$Pr(X = 0 $ or $ 1)$ for both odd and even is $2^{5} / 2^{10}$ = $\frac{1}{32}$.
$Pr(Y = 0) =$ $1$ subset {1,2} / $2^{10}$ possible subsets
$Pr$($X=0$ and $Y=0$) = $P$($X=0)P(Y=0)$
$\frac{1}{2^{10}}$ $.$ $\frac{1}{32}$. = $\frac{1}{32}$ $.$ $\frac{1}{2^{10}}$
They are equal and hence independent. Is my approach to finding the probabilities correct to come to this conclusion of independence?
Best Answer
The way to understand independence is to ask yourself: "If I know that variable $X$ has a particular value, does that provide any information whatsoever about the value of variable $Y$?"
a) So imagine $X = 7$. Does that provide any information whatsoever about the value of variable $Y$? Of course it does! You can be certain that $Y \neq 7$, for instance. (We can only have $Y=0$.)
b) Do the same analysis for your other cases.
Yes... you can also check this through:
Is $P(X|Y) = P(X)$.