Sunday, 8 June 2014

probability - Indicator Random Variables, Expectation, Covariance

[Not Homework]




Hey guys, I've been having trouble with this concept for a while as I just cant seem to get it down, can anyone help me with regards to this question I found in a paper?



Question



My understanding:
The expectation of a RV is just the probability of it happening, Hence,



for i)
$$
E[X_i] = P(X_i = 1) = P(r_i\hspace{0.1cm}is\hspace{0.1cm}withdrawn)

$$



However, since the balls are drawn without replacement, I'm unsure how as to get this probability (as ri would be first, or second etc) Hence, is it:



$$
\frac{1*\binom{29}{11}}{\binom{30}{12}}
$$



where 1 is fixing ball ri, and the rest is picking any balls at random?




for ii) and iii), I'm pretty much stumped. Can I assume their independence and thus,



$$
E[X_iY_i] = E[X_i]*E[Y_i]
$$



Thank you for your help and guidance! :D

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