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Let $H_{1}$, ..., $H_{n}$ be n events. $H_{1}$, ..., $H_{n}$ are jointly independent (or mutually independent) if and only if for any sub-collection of k events ($k \leq n$) $H{_{i1}}$, ..., $H_{i_{k}}$

My definition of mutual independence: $P(\cap^{k}_{j=1}E_{1j}) = \Pi^{k}_{j=1}P(E_{ij})$.

I read this and assumed something along the lines of:

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

Let $H_1$ be equal to the event of getting a heads on the first toss. So $H_1$ = {HHH, HHT, HTT, HTH}.

I then thought that an example of a sub-collection of an event would be $H_{11} = {HHH}$. Although, I think I am wrong.

But I do not see how my idea of a sub-collection of an event, helps clear up my definition of mutual independence. I think that I may be reading the definition wrong.

edit: $E_{12} = HHT, E_{13} = HTT, \ldots$

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1 Answer 1

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Let $H_1, H_2, H_3$ be the events we need to be independent. Any subcollection of events means that we can take any of these events. To be independent, they should satisfy four conditions: $$ k=2,\; i_1=1,\;i_2=2: \quad \mathbb P(H_1\cap H_2)=\mathbb P(H_1)\mathbb P(H_2). $$ $$ k=2,\; i_1=1,\;i_2=3: \quad \mathbb P(H_1\cap H_3)=\mathbb P(H_1)\mathbb P(H_3). $$ $$ k=2,\; i_1=2,\;i_2=3: \quad \mathbb P(H_2\cap H_3)=\mathbb P(H_2)\mathbb P(H_3). $$ $$ k=3,\; i_1=1,\;i_2=2,\;i_3=3: \quad \mathbb P(H_1\cap H_2\cap H_3)=\mathbb P(H_1)\mathbb P(H_2)\mathbb P(H_3). $$

If you take $H_1=\{HHH, HHT, HTT, HTH\}$, you should take the other events $H_2$, $H_3$ and so on to check independence. Say, $H_2=\{HHH, HHT, THH, THT\}$ means that the second coin is head. Let also $H_3=\{HHH, HTH, THH, TTH\}$ means that the third coin is head.

Take $k=2$ and events with numbers $i_1=1$ and $i_2=2$. They are $H_1$ and $H_2$. Their intersection is $H_1\cap H_2 = \{HHH, HHT\}$. Probability of this event is $\frac28=\frac14=\frac12\cdot\frac12=\mathbb P(H_1)\cdot \mathbb P(H_2)$.

Then take the other pari of events $H_1$ and $H_3$ and so on.

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