A biased coin with P(Head) = 0.4 is tossed 1000 times. Let X be the counts...

A biased coin with P(Head) = 0.4 is tossed 1000 times. Let X be the counts of heads in the tossing.

Find the upper bound for P(X ≤ 300 or X ≥ 500).
Use Gaussian approximation to compute P(X ≤ 300 or X ≥ 500).

Expert Solution

orchestra answered 2 years ago

A biased coin (P(head) = p , p is not equal to 0.5) is tossed repeatedly....

A biased coin (P(head) = p , p is not equal to 0.5) is tossed repeatedly. Find the probability that there is a run of r heads in a row before there is a run of s tails, where r and s are positive integers. Express the answer in terms of p, r and s.

a) A coin is tossed 4 times. Let X be the number of Heads on the...

a) A coin is tossed 4 times. Let X be the number of Heads on the first 3 tosses and Y be the number of Heads on the last three tossed. Find the joint probabilities pij = P(X = i, Y = j) for all relevant i and j. Find the marginal probabilities pi+ and p+j for all relevant i and j. b) Find the value of A that would make the function Af(x, y) a PDF. Where f(x, y)...

A coin with probability p>0 of turning up heads is tossed 4 times. Let X be...

A coin with probability p>0 of turning up heads is tossed 4 times. Let X be the number of times heads are tossed. (a) Find the probability function of X in terms of p. (b) The result above can be extended to the case of n independent tosses (that is, for a generic number of tosses), and the probability function in this case receives a very specific name. Find the name of this particular probability function. Notice that the probability...

A coin with P(head) = 2/3 is tossed six times (independently). What is the probability of...

A coin with P(head) = 2/3 is tossed six times (independently). What is the probability of obtaining exactly four CONSECUTIVE heads or exactly five CONSECUTIVE heads?

A coin is tossed 6 times. Let X be the number of Heads in the resulting...

A coin is tossed 6 times. Let X be the number of Heads in the resulting combination. Calculate the second moment of X. (A).Calculate the second moment of X (B). Find Var(X)

A fair coin will be tossed three times. (a) Indicating a head by H and a...

A fair coin will be tossed three times. (a) Indicating a head by H and a tail by T write down the outcome space. (b) What is the probability that on the first toss the outcome with a tail? (c) What is the probability of obtaining exactly two heads from the three coin tosses? (d) What is the probability that the first toss gives a tail and exactly two heads are obtained from the three coin tosses? Are the outcomes...

A fair coin is tossed four times. Let X denote the number of heads occurring and...

A fair coin is tossed four times. Let X denote the number of heads occurring and let Y denote the longest string of heads occurring. (i) determine the joint distribution of X and Y (ii) Find Cov(X,Y) and ρ(X,Y).

A fair coin is tossed 3 times. Let X be equal to 0 or 1 accordingly...

A fair coin is tossed 3 times. Let X be equal to 0 or 1 accordingly as head or tail occurs on a first Toss. Let y equal to number of heads that occurs. Find A) the distribution of X and Y B) the joint distribution of X and Y C) whether they are independent or not D) the COV (XY)

1. A coin is tossed 3 times. Let x be the random discrete variable representing the...

1. A coin is tossed 3 times. Let x be the random discrete variable representing the number of times tails comes up. a) Create a sample space for the event; b) Create a probability distribution table for the discrete variable x; c) Calculate the expected value for x. 2. For the data below, representing a sample of times (in minutes) students spend solving a certain Statistics problem, find P35, range, Q2 and IQR. 3.0, 3.2, 4.6, 5.2 3.2, 3.5...

An unbiased coin is tossed four times. Let the random variable X denote the greatest number...

An unbiased coin is tossed four times. Let the random variable X denote the greatest number of successive heads occurring in the four tosses (e.g. if HTHH occurs, then X = 2, but if TTHT occurs, then X = 1). Derive E(X) and Var(X). (ii) The random variable Y is the number of heads occurring in the four tosses. Find Cov(X,Y).

Question