# Question

Develop a careful proof of Theoram 2.1 which states that for any events A and B,

Pr (A U B) = Pr (A) + Pr (B) – Pr (A ∩ B).

One way to approach this proof is to start by showing that the set can be written as the union of three mutually exclusive sets,

And hence by Corollary 2.1,

Next, Show that

And likewise

Put these results together to complete the desired proof.

Pr (A U B) = Pr (A) + Pr (B) – Pr (A ∩ B).

One way to approach this proof is to start by showing that the set can be written as the union of three mutually exclusive sets,

And hence by Corollary 2.1,

Next, Show that

And likewise

Put these results together to complete the desired proof.

## Answer to relevant Questions

Show that the above formula for the probability of the union of two events can be generalized to three events as follows: Pr (A U B U C) = Pr (A) + Pr (B) + Pr(C) – Pr( A∩ B)– Pr( A∩ C)– Pr( B∩ C)+ Pr( A∩ B∩ ...A certain random variable has a probability density function of the form f x (x) = ce –2xu(x). Find the following: (a) The constant c, (b) Pr (X >2), (c) Pr (X < 3), (d) Pr (X < 3|X > 2). Suppose a random variable has a CDF given by Find the following quantities: Pr( X < 2) Pr( X > 4) Pr( 1< X< 3) Pr( X > 2|X < 4) The phase of a sinusoid, ϴ, is uniformly distributed over so that its PDF is of the form (a) (b) (c) Let be a Gaussian random variable such that X ~ N (0, σ2). Find and plot the following conditional PDFs. (a) fx|x > 0 (x) (b) fx||x > 3 (x) (c) fx||x >3 (x)Post your question

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