Let X be the number of heads in the first n flips and Y be the number of heads in the next m flips.
(a) Are X and Y independent?
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...
Let X equal to the number of heads after 4 flips of a fair coin? Derive the probability mass function for X, and plot it. Also, compute the E[X] of X.
A fair coin is tossed n times. Let X be the number of heads in this n toss. Given X = x, we generate a Poisson random variable Y with mean x. Find Var[Y]. Answer depends on n.
The PMF of the experiment that records the number of heads in four flips of a coin, which can be obtained with the R commands attach (expand.grid (X1=0:1, X2=0:1, X3=0:1, X4=0:1)); table(X1+X2+X3+X4)/length(X1), is x 0 1 2 3 4 p(x) 0.0625 0.25 0.375 0.25 0.0625 Thus, if the random variable X denotes the number of heads in four flips of a coin then the probability of, for example, two heads is P(X = 2) = p(2) = 0.375. What is...
Let X be the number of heads in n tosses of a fair coin. For each of the expected value by 10 or less. Use the continuity correction. Notice that these probabilities decrease as n increases. This may seem to contradict the fact 9. following values of n, find the approximate probability that X differs from its a. 100 b. n 500 c.n 1,500 d.n 5,000 t X, the frequency of heads, is supposed to be close to n/2 when...
Question 3. Define X as the number of heads observed in an experiment that flips a balanced coin 3 times. Calculate: a) The expected value of X. b) The probability that X 2 c) The probability that X22
A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K = 5. For k 1, 2, , K, let Xk be a continuous random variable that is uniform over the interval [0, 5]. The Xk are independent of one another and of the coin flips. LetX = Σ i Xo Find the...
A fair coin is tossed 20 times. Let X be the number of heads thrown in the first 10 tosses, and let Y be the number of heads tossed in the last 10 tosses. Find the conditional probability that X = 6, given that X + Y = 10.
A fair coin is tossed 20 times. Let X be the number of heads thrown in the first 10 tosses, and let Y be the number of heads tossed in the last 10 tosses. Find the conditional probability that X = 6, given that X + Y = 10.
Problem 4. Five coins are flipped. The first four coins will land on heads with probability 1/4. The fifth coin is a fair coin. Assume that the results of the flips are independent. Let X be the total number of heads that result Hint: Condition on the last flip. (a) Find P(X2) (b) Determine E[X] S.20