Bernoulli Random variable and population problems - i need it for an exam

Let Y_{a }and Y_{b} denote Bernoulli random variables from 2 different populations, denoted a and b. suppose that E (Y_{a})= P_{a }and E (Y_{b}) = P_{b}. A random sample of size n_{a} is chosen from population a, with sample average denoted P^_{a} and a random sample of size n_{b} is chosen from the population b, with sample average denoted p^_{b} Sample from population a is independent from sample from population b.

Show that E (P^_{a}) = P_{a }and var (P^_{a})= P_{a}(1- P_{a})/ n_{a.}

Show that E(P^_{b}) = P_{b }and var (P^_{b})= P_{b}(1- P_{b})/ n_{b.}

Show that var (P^_{a }- P^_{b})= P_{a}(1- P_{a})/ n_{a} + P_{b}(1- P_{b})/ n_{b}

Suppose that n_{a }and n_{b} are large. Show that a 95% confidence interval for P_{a }- P_{b }is given by (P^_{a }- P^_{b}) plus and minus 1.96 sqrt P^_{a}(1- P^_{a})/ n_{a} + P^_{b}(1- P^_{b})/ n_{b.}

How would you construct a 90% confidence interval for P_{a }- P_{b}?

I have missed the class when prof. taught these.... :(

Re: Bernoulli Random variable and population problems - i need it for an exam

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**homalina** Let Y_{a }and Y_{b} denote Bernoulli random variables from 2 different populations, denoted a and b. suppose that E (Y_{a})= P_{a }and E (Y_{b}) = P_{b}. A random sample of size n_{a} is chosen from population a, with sample average denoted P^_{a} and a random sample of size n_{b} is chosen from the population b, with sample average denoted p^_{b} Sample from population a is independent from sample from population b.

Show that E (P^_{a}) = P_{a }and var (P^_{a})= P_{a}(1- P_{a})/ n_{a.}

Show that E(P^_{b}) = P_{b }and var (P^_{b})= P_{b}(1- P_{b})/ n_{b.}

Show that var (P^_{a }- P^_{b})= P_{a}(1- P_{a})/ n_{a} + P_{b}(1- P_{b})/ n_{b}

Suppose that n_{a }and n_{b} are large. Show that a 95% confidence interval for P_{a }- P_{b }is given by (P^_{a }- P^_{b}) plus and minus 1.96 sqrt P^_{a}(1- P^_{a})/ n_{a} + P^_{b}(1- P^_{b})/ n_{b.}

How would you construct a 90% confidence interval for P_{a }- P_{b}?

I have missed the class when prof. taught these.... :(

I have given you hints on finding the expected value and variance of sample average in one of your previous problems..use the same concept here