2. Calculate E[X].
2. The MAP rule decides in favor of Heads if X<a and in favor of Tails if X>a. What is a?
2. Calculate E[X]. 2. The MAP rule decides in favor of Heads if X<a and in...
You flip a coin 100 times. Let X= the number of heads in 100 flips. Assume we don’t know the probability, p, the coin lands on heads (we don’t know its a fair coin). So, let Y be distributed uniformly on the interval [0,1]. Assume the value of Y = the probability that the coin lands on heads. So, we are given Y is uniformly distributed on [0,1] and X given Y=p is binomially distributed on (100,p). Find E(X) and...
Q3. Suppose we toss a coin until we see a heads, and let X be the number of tosses. Recall that this is what we called the geometric distribution. Assume that it is a fair coin (equal probability of heads and tails). What is the p.m.f. of X? (I.e., for an integer i, what is P(X=i)? What is ?[X]? ({} this is a discrete variable that takes infinitely many values.)
9.74. Suppose we toss a biased coin independently until we get two heads or two tails in total. The coin produces a head with probability p on any toss. 1. What is the sample space of this experiment? 2. What is the probability function? 3. What is the probability that the experiment stops with two heads?
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 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 box contains five coins. For each coin there is a different probability that a head will be obtained when the coin is tossed. (Some of the coins are not fair coins!) Let pi denote the probability of a head when the i th coin is tossed (i = 1, . . . , 5), and suppose that p1 = 0, p2 =1/4, p3 =1/2, p4 =3/4, p5 =1. The experiment we are interested in consists in selecting at random...
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...
2. Let X be the number of Heads when we toss a coin 3 times. Find the probability distribution (that is, the probability function) for X.
Question 2 Suppose you have a fair coin (a coin is considered fair if there is an equal probability of being heads or tails after a flip). In other words, each coin flip i follows an independent Bernoulli distribution X Ber(1/2). Define the random variable X, as: i if coin flip i results in heads 10 if coin flip i results in tails a. Suppose you flip the coin n = 10 times. Define the number of heads you observe...
2. Suppose we want to test whether a coin is fair (that is, the probability of heads is p = .5). We toss the coin 1000 times, and record the number of heads. Let T denote the number of heads divided by 1000. Consider a test that rejects the null hypothesis that p=.5 if T > c. (a) Write down a formula for P(T>c) assuming p = 0.5. (This formula may be compli- cated, but try to give an explicit...