Consider a statistical expirement of flipping a pair of fair coins simultaneously. Let X & Y...

Consider a statistical expirement of flipping a pair of fair coins simultaneously. Let X & Y be the number heads in flipping each coin.

Define a joint density function Z = XY. (I provided the answers in bold but I need help understanding how it is solved. Thank you in advance).

a) The number of possible distinct values of Z is: 2

b) The probability of Z = 0 is: 3/4

c) The mean of Z is: 1/4

d) The standard deviation of Z is: sqrt(3/16)

e) The correlation coefficient of X and Y is: 0

f) Which RV follows a uniform distribution among all three random variables, X, Y, and Z? X & Y

Solutions

Expert Solution

f. If all values of random variable have same probability then that random variable has uniform distribution.

X and y having equal probability for its all values but z does not.

Hence X and Y follow uniform distribution.

orchestra answered 1 year ago

Next > < Previous

Related Solutions

4 fair coins are tossed. Let X be the number of heads and Y be the...

4 fair coins are tossed. Let X be the number of heads and Y be the number of tails. Find Var(X-Y) Solution: 3.5 Why?

Consider independent trials of flipping fair coins (outcomes are heads or tails). Define the random variable...

Consider independent trials of flipping fair coins (outcomes are heads or tails). Define the random variable T to be the first time that two heads come up in a row (so, for the outcome HT HT HH... we have T = 6). (a) Compute P(T = i) for i = 1, 2, 3, 4, 5. (b) Compute P(T = n) for n > 5.

Seven fair coins are flipped. The outcomes are assumed to be independent. Let X be the...

Seven fair coins are flipped. The outcomes are assumed to be independent. Let X be the number of heads. What is the probability that X < 3? What is the probability that X ≥ 4? What is the probability that 3 ≤ X < 7

Consider rolling two dice and let (X, Y) be the random variable pair defined such that...

Consider rolling two dice and let (X, Y) be the random variable pair defined such that X is the sum of the rolls and Y is the maximum of the rolls. Find the following: (1) E[X/Y] (2) P(X > Y ) (3) P(X = 7) (4) P(Y ≤ 4) (5) P(X = 7, Y = 4)

Three fair coins are flipped independently. Let X be the number of heads among the three...

Three fair coins are flipped independently. Let X be the number of heads among the three coins. (1) Write down all possible values that X can take. (2) Construct the probability mass function of X. (3) What is the probability that we observe two or more heads. (i.e., P(X ≥ 2)) (4) Compute E[X] and Var(X).

Let X be the random variable for the number of heads obtained when three fair coins...

Let X be the random variable for the number of heads obtained when three fair coins are tossed: (1) What is the probability function? (2) What is the mean? (3) What is the variance? (4) What is the mode?

1. Suppose four distinct, fair coins are tossed. Let the random variable X be the number...

1. Suppose four distinct, fair coins are tossed. Let the random variable X be the number of heads. Write the probability mass function f(x). Graph f(x). 2. For the probability mass function obtained, what is the cumulative distribution function F(x)? Graph F(x). 3. Find the mean (expected value) of the random variable X given in part 1 4. Find the variance of the random variable X given in part 1.

Assume we are flipping 1000 fair coins. What can you say about the probability that you...

Assume we are flipping 1000 fair coins. What can you say about the probability that you will get at least 3/5 of them to be heads? Compare the 3 bounds (Markov, Chebyshev, Chernoff), and try plugging in some leargevalue of n to see how muchs they differ!

Two rolls of a fair die. Let x and y be the results of the two...

Two rolls of a fair die. Let x and y be the results of the two rolls, and let z = x + y. (a) Find P[x = 4, y = 3], P[x > 3, y = 5]. (b) Find P[z = 7], P[z = 5], P[z = 3]. (c) Find the probability that at least one 6 appears given that z = 8. (d) Find the probability that x = 6 given that z > 8. (e) Find the...

Three fair coins are tossed simultaneously. We define the events A: get at least two heads...

Three fair coins are tossed simultaneously. We define the events A: get at least two heads B: get two tails C: get at most two heads i) Write down the sample space S. ii) Calculate P(A), P(B), P(C). iii) Calculate P(A ∩ B), P(A ∩ C), P(B ∩ C). iv) Calculate P(A Ս B), P(A Ս C), P(B Ս C). v) Which of the following pairs of events are independent and why? A and B A and C vi) Which of the following pairs of events...

Subjects