A coin has a probability x of landing heads and 1-x of landing tails, where x has a value between 0 and 1. Prove that the SMI of the coin toss is maximized when x = 1/2.
* Edit: I'm not sure what SMI is, maybe Shannon Mutual Information?
A coin has a probability x of landing heads and 1-x of landing tails, where x has a value between...
A coin has a probability x of landing heads and 1-x of landing tails, where x has a value between 0 and 1. Prove that the SMI of the coin toss is maximized when x = 1/2. * Edit: I'm not sure what SMI is, maybe Shannon Mutual Information?
Consider a coin with probability q of landing on heads, and probability 1−q of landing on tails. a) The coin is tossed N times. What is the probability that the coin lands k times on heads. b) The coin is tossed 100 times, and lands on heads 70 times. What is the maximum likelihood estimate for q?
Problem 2: Tails and (Heads or Tails?) Alice and Bob play a coin-tossing game. A fair coin (that is a coin with equal probability of 1. The coin lands 'tails-tails' (that is, a tails is immediately followed by a tails) for the first 2. The coin lands 'tails-heads (that is, a tails is immediately followed by a heads) for the landing heads and tails) is tossed repeatedly until one of the following happens time. In this case Alice wins. first...
The Belgian Euro coin is known to be biased: it has a probability of 0.56 of landing on heads when flipped, and a probability of 0.44 of landing on tails. Answer the questions below using the event ‘landing on heads’ as a success, and ‘landing on tails’ as a failure. 1. What is the expected value for heads of flipping the Belgian Euro coin 50 times? 2. What is the standard deviation for flipping the Belgian Euro coin 50 times?
2. SUPPLEMENTAL QUESTION 1 (a) Toss a fair coin so that with probability pheads occurs and with probability p tails occurs. Let X be the number of heads and Y be the number of tails. Prove X and Y are dependent (b) Now, toss the same coin n times, where n is a random integer with Poisson distribution: n~Poisson(A) Let X be the random variable counting the number of heads, Y the random variable counting the number of tails. Prove...
4. Toss a fair coin 6 times and let X denote the number of heads that appear. Compute P(X ≤ 4). If the coin has probability p of landing heads, compute P(X ≤ 3) 4. Toss a fair coin 6 times and let X denote the number of heads that appear. Compute P(X 4). If the coin has probability p of landing heads, compute P(X < 3).
An unfair coin has probability 0.4 of landing heads. The coin is tossed seven times. What is the probability that it lands heads at least once? Round your answer to four decimal places. P (Lands heads at least once) -
Assume that a coin is flipped where the probability of coin lands "Heads" is 0.49. The coin is flipped once more. This time, the probability of obtaining the first flip's result is 0.38. The random variable X is defined as the total number of heads observed in two flips. On the other hand, the random variable Y is defined as the absolute difference between the total number of heads and the total number of tails observed in two flips. Calculate...
All work must be shown 20. An unfair coin has probability 0.65 of landing tails. The coin is tossed three times. What is the probability that it lands tails at least once? Express your answer as a decimal.
Let X represent the number of heads subtracts the number of tails obtained when a coin is tossed 3 times, i.e., X = number of heads − number of tails. (a) Find the probability mass function of X (b) Given that X is at least 0, what is the probability that X is at least 2