Let X be a Binomial random variable defined as the sum of 6 independent Bernoulli trials. The probability of a Bernoulli taking the value 1 is given by p. Suppose that prior to the 6 Bernoulli trials, p is chosen to take one of three possible values with the following probabilities:
p Probability
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0.2 0.1
0.6 0.7
0.8 0.2
• Compute the joint probability distribution of X and p. Are X and p independent?
• Compute the unconditional mean and variance of X.
• Compute the conditional mean of X given each possible value of p. Based on your calculations, what sign do you expect the covariance between X and p to be?
I'm having trouble understanding the lead-up of the question. I'm interpreting it that the probability of X is dependent on p, which also has probabilities and this is confusing me. What would the sample space of X be and how could you find the probabilities of X being values other than 1? Also for the probability of X being 1, how do you know which p value to use (0.2, 0.6, 0.8 from the table)? Thanking you in advance!
Best Answer
The joint distribution of the random variables $X$ and $P$ is given by $$pr(X=x,P=p) = pr(X=x\ |\ P=p) \ pr(P=p),$$ where $$pr(X=x\ |\ P=p) = \binom{6}{x}p^x (1-p)^{6-x}\ (x=0,1,2,3,4,5,6) $$ and $pr(P=p)$ is given by your table, for $p=0.2, 0.6, 0.8$.
The unconditional mean and variance of $X$ is found from the marginal distribution $$pr(X=x) = \sum_{p} pr(X=x,P=p). $$
This should be enough to answer all of the stated questions.