Assume a die is rolled 100 times. Let Xư be the result of k-th roll and...
3. A die is to be rolled 90 times. Let X count the number of times that we roll a 5. Determine E[X]
A fair die is rolled 100 times. Let X add the faces of all of the rolls together. Then µ = 350. Find an upper bound for P(X ≥ 400). Find the actual probability P(X = 100)
Use a random number generator to simulate the roll of a fair die 100 times. Let the number face up on the die represent the variable X A. Build a relative frequency table of the outcomes of the variable X. X Freq Rel. Freq B. Use the relative frequency distribution from part c to estimate the probability of an even number face up, then find the actual probability using the probability distribution and comment on the difference in values.
5. Roll the die another 40 times and calculate the value of x. Sample Mean Observation (= second observation of X): 6. Now write your two X values (one from question 2 and one from question 5). Comment on the values. 7. The random variable X represents the outcome of a single roll of the die, and the random variable X represents the sample mean of 40 rolls of the die. Use the Central Limit Theorem, and the values in...
I know Pk~1/k^5/2 just need the
work
Problem 1. Suppose that a fair six-sided die is rolled n times. Let N be the number of 1's rolled, N2 be the number of 2's rolled, etc, so that NN2+Ns-n Since the dice rolls are independent then the random vector < N,, ,Ne > has a multinomial distribution, which you could look up in any probability textbook or on the web. If n 6k is a multiple of 6, let Pa be...
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...
2. Assume two fair dice are rolled. Let X be the number showing on the first die and number showing on the second die. (a) Construct the matrix showing the joint probability mass function of the pair X,Y. (b) The pairs inside the matrix corresponding to a fixed value of X - Y form a straight line of entries inside the matrix. Draw those lines and use them to construct the probability mass function of the random variable X-Y- make...
An unfair coin has probability p = 0.20 of being heads and is flipped 100 times. (a) What is the exact distribution, mean, and variance of X the number of heads that appears in this experiments? (b) What is the exact probability that between 15 and 28 heads appear? Please express your solution as a summation. (c) Using the Central Limit Theorem, what is the approximate probability that between 15 and 28 heads appear? Please simplify your solution. (Do not...
Question 3: Let X1,..., X.be iid Poisson (2) random variables. a. Find the maximum likelihood estimate for X. b. Obtain the Fisher expected information. c. Obtain the observed information evaluated at the maximum likelihood estimate. d. For large n, obtain a 95% confidence interval for based on the Central Limit Theorem. e. Repeat part (a), but use the Wald method. f. Repeat part (d), but use the Score method. 8. Repeat part (a), but use the likelihood ratio method.
Let X1, X2, ..., X48 denote a random sample of size n = 48 from the uniform distribution U(?1,1) with pdf f(x) = 1/2, ?1 < x < 1. E(X) = 0, Var(X) = 1/3 Let Y = (Summation)48, i=1 Xi and X= 1/48 (Summation)48, i=1 Xi. Use the Central Limit Theorem to approximate the following probability. 1. P(1.2<Y<4) 2. P(X< 1/12)