Exercise 4.9. Let X ~ Poisson(10). (a) Find P(X>7). (b) Find P(X < 13 X >...
5. (10 points) Let p="x < y", q="x < 1", and r="y > 0". Using ~, 1, V write the following statements in terms of the symbols p, q, and r. (a) 0 <y < x < 1. (b) 1 < x <y<0.
Exercise 7. Let X be a standard normal random variable. Prove that for any integer n > 0, ELY?"] = 1207) and E[x2n+] = 0.
a b and c please
and thankyou
Problem 2 Let X is a random variable with Poisson distribution X ~ Poisson(λ), (a) Find E(X1X2 i). λ > 0. 、 (b) Find E(xIx2). (c)Prove that λ>2-2a-ka for λ>0.
Problem 3: Conditional Probabilities Let A and B be events. Show that P(An B | B) = P(A | B), assuming P(B) > 0.
Let X be a discrete random variable that follows a Poisson distribution with = 5. What is P(X< 4X > 2) ? Round your answer to at least 3 decimal places. Number
10) The X random variable has a normal distribution. P(X > 15) = 0.0082 and P(X<5) = 0.6554 find the mean and variance of this distribution
Exercise 1.25. This exercise relates to (1.13). Suppose that x > 1. For each ne N let y= Vå be defined by yn = x. This implies that vx < mă if n > m. To prove that Veso 3NEN Vnən : 0< V2-1<e it therefore suffices to prove that Vesonen: 0< Vx-1<e. Prove this latter statement.
Exercise 6.14 Let y be distributed Bernoulli P(y = 1) unknown 0<p<1 p and P(y = 0) = 1-p f or Some (a) Show that p E( (b) Write down the natural moment estimator p of . (c) Find var (p) (d) Find the asymptotic distribution of vn (-p) as no. as n> OO.
Question 12: Let X and Y have the joint probability density function Find P(X>Y), P(X Y <1), and P(X < 0.5)
Let descrete random variable X ~ Poisson(7). Find: 1) Probability P(X = 8) 2) Probability P(X = 3) 3) Probability P(X<4) 4) Probability P(X> 7) 5) ux 6) 0x Show your explanations. Displaying only the final answer is not enough to get credit. Note: round calculated numerical values to the fourth decimal place where applicable.