Question 3 Suppose that the random variable X has the Poisson distribution, with P (X0) 0.4....
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and further suppose that U is a uniformly distributed random variable between 0 and 1 that is independent from X and Y. Calculate Pr(X<U< Y) and estimate numerically (based on a visual plot, for example) the value of λ that maximizes this probability.
Poisson Distribution Question Problem 2: Let the random variable X be the number of goals scored in a soccer game, and assume it follows Poisson distribution with parameter λ 2, t 1, i.e. X-Poisson(λ-2, t Recall that the PMF of the Poisson distribution is P(X -x) - 1) e-dt(at)*x-0,1,2,.. x! a) Determine the probability that no goals are scored in the game b) Determine the probability that at least 3 goals are scored in the game. c) Consider the event...
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~ Exp(A), so that its probability density function (pdf) of X is SO e /A fx(x) | 0, (2) (a) Let Y log(X. When A = 1, (i) Show that the pdf of Y is fr(y) = e (u+e-") (ii) Derive the moment generating function of Y, My(t), and give the values of t such that My(t) is well defined. (b) Suppose that Xi, i...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable X follows a Poisson distribution with parameter λ if Pr(X = k) = Ake-A ke(0, 1,2, ) k! You are a warrior in Peter Jackson's The Hobbit: Battle of the Five Armies. Because Peter decided to make his battle scenes as legendary as possible, he's decided that the number of orcs that will die with one swing of your sword is Poisson distributed (lid)...
Exercise 2.23 If X is a discrete random variable having the Poisson distribution with parameter that the probability that X is even is e cosh A. Exercise 2.24 If X is a discrete random variable having the geometric distribution with parameter p. show that the probability that X is greater than k is (1 -p)k à, show
Problem 6. [Poisson is Pronounced 'Pwah-ssohn] (a) Suppose that X is a random variable following the Poisson distribution with rate parameter A. Show that E[x]-A Hint: You may find the following fact useful: at k! (b) Suppose that we obtained the following count data: Count Frequency 24 30 17 19 Fit a Poisson distribution to the data using the Method of Moments (c) Suppose that X is a random variable that follows the Poisson distribution that you fit in part...
A discrete random variable X follows the geometric distribution with parameter p, written X ∼ Geom(p), if its distribution function is A discrete random variable X follows the geometric distribution with parameter p, written X Geom(p), if its distribution function is 1x(z) = p(1-P)"-1, ze(1, 2, 3, ). The Geometric distribution is used to model the number of flips needed before a coin with probability p of showing Heads actually shows Heads. a) Show that fx(x) is indeed a probability...
11. If the random variable X has a Poisson distribution such that P(X=1) = P(x=3), find P(X= 5). Give the answer with 6 dec. places
Let X be a discrete random variable that follows a Poisson distribution with λ=3. What is P(X<5|X>3)? Give your response to at least 3 decimal places.