Suppose that a dog can successfully shake a paw with independent and constant probability p. In a...
Suppose that a dog can successfully shake a paw with independent
and constant probability p. In a particular training session, let X
be the number of paw shakes until the dogs first failure. Let Y be
the number of paw shakes until the dogs second failure. Show
that
P(Y=2X) = (1-p) / (1+p)
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Suppose that a dog can successfully shake a paw with independent and constant probability p. In a...
Problem 5 (10 points). Suppose that the independent Bernoulli trials each with success probability p, are...
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).
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that...
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that there is no three consecutive successes in n trails. (1). Show that limn+cQn = 0 (2). Show that Qn = (1 - pQn-1 + p(1 - pQn-2 + p (1 - p)Qn-3 for n 3 (Hint: condition on the first failure). Problem 5. Suppose we do n independent trails that each has a probability P E (0,1) to result in success. Let Pn be...
You perform a sequence of m+n independent Bernoulli trials with success probability p between (0, 1)....
You perform a sequence of m+n independent Bernoulli trials with success probability p between (0, 1). Let X denote the number of successes in the first m trials and Y be the number of successes in the last n trials. Find f(x|z) = P(X = x|X + Y = z). Show that this function of x, which will not depend on p, is a pmf in x with integer values in [max(0, z - n), min(z,m)]. Hint: the intersection of...
probability course 01) 6 and Let X and Y be two independent random variables. Suppose that...
probability course
01) 6 and Let X and Y be two independent random variables. Suppose that we know Var(2X-Y) Var(X+ 2Y) 9, Find Var(X) and Var(Y).
3Y 2 1. (20 points) Suppose that X and Y independent random variables. Let W 2x (a) Consider the following probability...
3Y 2 1. (20 points) Suppose that X and Y independent random variables. Let W 2x (a) Consider the following probability distribution of a discrete random variable X: 12 P(X) 00.7 0.3 X Compute the mean and variance of X (b) Use your answers in part (a). If E(Y)=-3 and V(Y)= 1, what are E(W) and V (W)?
Problem 8. Suppose that XGeom(p) and Y ~ Geom(r) are independent. Find the probability P(X <Y).
Problem 8. Suppose that XGeom(p) and Y ~ Geom(r) are independent. Find the probability P(X <Y).
Suppose that total 5 independent trials having a common probability of success 1/3 are performed. If...
Suppose that total 5 independent trials having a common probability of success 1/3 are performed. If X is the number of successes in the first2 trials, and Y is the number of successes in the final 3 trials, then X and Y are independent, since knowing the number of successes in the first 2 trials does not affect the distribution of the number of successes in the final 3 trials (by the assumption of independent trials). Find the joint p.d.f....
1. (6) Let X be a random variable on probability space (2, F, P), and Y...
1. (6) Let X be a random variable on probability space (2, F, P), and Y X +1. Show that if x and Y are independent, then X is a constant with probability one.
question1: Suppose A, B & C are independent events with common probability = .20 Determine P(A...
question1: Suppose A, B & C are independent events with common probability = .20 Determine P(A U B U C) question2: A coin with P(heads) = p is tossed until heads appears. Determine the probability it takes an odd number of tosses.
2. Suppose that (X,Y) has the following joint probability density function: f(x,y) = C if -1...
2. Suppose that (X,Y) has the following joint probability density function: f(x,y) = C if -1 <r< 1 and -1 <y<1, and 0 otherwise. Here is a constant. (a) Determine the value of C. (b) Are X and Y independent? (Explain why or why not.) (c) Calculate the probability that 2X - Y > 0 (d) Calculate the probability that |X+Y| < 2 3. Suppose that X1 and X2 are independent and each is standard uniform on (0,1]. Let Y...
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).
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that there is no three consecutive successes in n trails. (1). Show that limn+cQn = 0 (2). Show that Qn = (1 - pQn-1 + p(1 - pQn-2 + p (1 - p)Qn-3 for n 3 (Hint: condition on the first failure). Problem 5. Suppose we do n independent trails that each has a probability P E (0,1) to result in success. Let Pn be...
probability course
01) 6 and Let X and Y be two independent random variables. Suppose that we know Var(2X-Y) Var(X+ 2Y) 9, Find Var(X) and Var(Y).
3Y 2 1. (20 points) Suppose that X and Y independent random variables. Let W 2x (a) Consider the following probability distribution of a discrete random variable X: 12 P(X) 00.7 0.3 X Compute the mean and variance of X (b) Use your answers in part (a). If E(Y)=-3 and V(Y)= 1, what are E(W) and V (W)?
Problem 8. Suppose that XGeom(p) and Y ~ Geom(r) are independent. Find the probability P(X <Y).
Suppose that total 5 independent trials having a common probability of success 1/3 are performed. If X is the number of successes in the first2 trials, and Y is the number of successes in the final 3 trials, then X and Y are independent, since knowing the number of successes in the first 2 trials does not affect the distribution of the number of successes in the final 3 trials (by the assumption of independent trials). Find the joint p.d.f....
1. (6) Let X be a random variable on probability space (2, F, P), and Y X +1. Show that if x and Y are independent, then X is a constant with probability one.
2. Suppose that (X,Y) has the following joint probability density function: f(x,y) = C if -1 <r< 1 and -1 <y<1, and 0 otherwise. Here is a constant. (a) Determine the value of C. (b) Are X and Y independent? (Explain why or why not.) (c) Calculate the probability that 2X - Y > 0 (d) Calculate the probability that |X+Y| < 2 3. Suppose that X1 and X2 are independent and each is standard uniform on (0,1]. Let Y...
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