Independent trials, each of which is a success with probability
p, are successively performed. Let X denote the first trial
resulting in a success. That is, X will equal k if the first k −1
trials are all failures and the kth a success. X is called a
Geometric random variable (google it). Determine the moment
generating function of X.
. In a sequence of 7 Bernoulli trials with probability of
success p, let X be the number of successes not followed
immediately by a failure. Find E(X) (you can use indicators)
Shown below are the number of trials and success probability for
some Bernoulli trials. Let X denote the total number of
successes.
n = 6 and p = 0.3
Determine P(x=4) using the binomial probability formula.
b. Determine P(X=4) using a table of binomial
probabilities.
Compare this answer to part (a).
Given the number of trials, n, and the probability of success,
p, find the following probabilities.
a. N = 12, p = 0.06 P(3 successes)
b. N = 18, p = 0.96 P(2 failures)
c. N = 5, p = 0.28 P(at least 3 successes)
d. N = 90, p = 0.04 P(no more than 4 successes)
9.8 Let X and Y be independent random variables with probability
distributions given by
P(X = 0) = P(X = 1) = 1/2 and P(Y = 0) = P(Y = 2) = 1/2 .
a. Compute the distribution of Z = X + Y .
b. Let Y˜ and Z˜ be independent random variables, where Y˜ has
the same distribution as Y , and Z˜ the same distribution as Z.
Compute the distribution of X˜ = Z˜ − Y
Let N be a binomial random variable with n = 2 trials and
success probability p = 0.5. Let X and Y be uniform random
variables on [0, 1] and that X, Y, N are mutually independent. Find
the probability density function for Z = NXY . Hint: Find P(Z ≤ z)
for z ∈ [0, 1] by conditioning on the value of N ∈ {0, 1, 2}.
Negative Binomial experiment is based on sequences of Bernoulli
trials with probability of success p. Let x+m be the number of
trials to achieve m successes, and then x has a negative binomial
distribution. In summary, negative binomial distribution has the
following properties
Each trial can result in just two possible outcomes. One is
called a success and the other is called a failure.
The trials are independent
The probability of success, denoted by p, is the same on every...
Let X and Y be i.i.d. geometric random variables with parameter
(probability of success) p, 0 < p < 1. (a) (6pts) Find P(X
> Y ). (b) (8pts) Find P(X + Y = n) and P(X = k∣X + Y = n), for
n = 2, 3, ..., and k = 1, 2, ..., n − 1
Let X and Y be i.i.d. geometric random variables with parameter
(probability of success) p, 0<p<1.
(a) Find P(X>Y).
(b) Find P(X+Y=n) and P(X=k|X+Y=n), for n=2,3,..., and
k=1,2,...,n−1.
I need an answer asap please. Thank you.
A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of he experiment. n=9, p=0.3, x≤3The probabity of x≤3 succenses is _______ (Round to four decimal places as needed.)