A geometric distribution is a series of independent Bernoulli trials.
True or false?
A geometric distribution is a series of independent Bernoulli trials. True or false?
Problem 5 (10 points). Suppose that the independent Bernoulli trials each with success probability p, are performed independently until the first success occurs, Let Y be the number of trials that are failure. (1) Find the possible values of Y and the probability mass function of Y. (2) Use the relationship between Y and the random variable with a geometric distribution with parameter p to find E(Y) and Var(Y).
The geometric distribution is a probability distribution of the number X of Bernoulli trials needed to get one success. For example, how many attempts does a basketball player need to get a goal. Given the probability of success in a single trial is p, the probability that the xth trial is the first success is: Pr(x = x|p) = (1 - p*-'p for x=1,2,3,.... Suppose, you observe n basketball players trying to score and record the number of attempts required...
5. A series of independent Bernoulli(p) trials is performed, labeled 1,2,3,...Let X be the location (label) of the first success observed, and Y be the location of the second success observed. (a) Find the joint PMF of X and Y; give your result as a general formula. (It may help to start by considering specific sequences, such as "001001", where X 3 and 6 (b) Write out thefirst of your PMF in table format, covering the range 1 sX 4,13YS...
Suppose Xi, X2,.. are independent Geometric (number of trials) random variables where ~Geometric p 1- a) It is easily shown that Xfor some constant a. Name it. a= b) According to the Borel-Cantelli Lemmas, does a.s /n In other words, will there eventually reach a point in the sequence of random variables where every X a? Suppose Xi, X2,.. are independent Geometric (number of trials) random variables where ~Geometric p 1- a) It is easily shown that Xfor some constant...
You are performing 7 independent Bernoulli trials with p = 0.2 and q = 0.8. Calculate the probability of the stated outcome. Check your answer using technology. (Round your answer to five decimal places.) P(X≥3) = _______
Suppose X1, X2,... are independent Geometric (number of trials) random variables where Xi ~ Geometric(p = 1/i^2) a) It is easily shown that Xn converges to a for some constant a. Name it. b) According to the Borel-Cantelli Lemmas, does Xn almost surely converge to a? Suppose Xi, X2, are independent Geometric (number of trials) random variables where x,~ Geometric(pal+) |. a) It is easily shown that Xa for some constant a. Name it. b) According to the Borel-Cantelli Lemmas,...
An experiment consists of 9 independent Bernoulli trials, each with a success probabilityof 0.6 Find P(7 <= X <= 9)
Suppose X1,X2,…,Xn represent the outcomes of n independent Bernoulli trials, each with success probability p. Note that we can write the Bernoulli distribution as: Suppose X1 2 X, represent the outcomes of n independent Bernou i als, each with success probabil ,p. Note that we can writ e the Bernoulǐ distribution as 0,1 otherwise Given the Bernoulli distributional family and the iid sample of X,'s, the likelihood function is: -1 a. Find an expression for p, the MLE of p...
You are performing 6 independent Bernoulli trials with p = 0.2 and q = 0.8. Calculate the probability of the stated outcome. Check your answer using technology. (Round your answer to four decimal places.) No successes
Assume a sequence of independent trials, each with probability p of success. Use the Law of Large Numbers to show that the proportion of successes approaches p as the number of trials becomes large. It may be useful to think of this problem as a Bernoulli distribution and to then calculate the mean.