ny? Consider a probability experiment in which the sample space is S = {1,2} How many events are there in this probability experiment? {1}{2}{1,2}
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- Give an example of a random variable X : {b, c, d, e} → N (Natural Number) with expectation 2, where each of {b, c, d, e} has equal probabilitySelect the events Consider a probability experiment with the sample space S = {1,2, 3} Select all but only the events for this experiment. O 2 O 1,2 O {1} O {} O 1,2 O {1,2}Probability Consider a probability experiment with S = {0, 1, 2, 3, 4, 5, 6, 7} and these two events: A = {1,2, 3}, and B= {1,3, 5, 6, 7} What is the probability that A occurs? What is the probability that B occurs? What is the obability that A n B occurs? What is the probability that AU Boccurs? What is the probability that {} occurs? What is the probability that S occurs?
- SIMULATION AND MODELING Using the mid- square method obtain the random variables using Z0= 1009 until the cycle degenerates to zero.2. Given a Sample Space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, Event A = {1, 3, 4, 7, 9}, and Event B = {3, 7, 9, 11, 12, 13} Find the probability P(A|B). State your answer as a value with one digit after the decimal point.If the probability of an event is 62%, what is the probability of its complement?
- Using R codes 3.6 Let X be a random variable with pmf given by p(x) = 1/10 for a = 1,..., 10 and p(x) = 0 for all other values of x. Find E[X] and confirm your answer using a simulation.Use R to answer the following question According to the central limit theorem, the sum of n independent identically distributed random variables will start to resemble a normal distribution as n grows large. The mean of the resulting distribution will be n times the mean of the summands, and the variance n times the variance of the summands. Demonstrate this property using Monte Carlo simulation. Over 10,000 trials, take the sum of 100 uniform random variables (with min=0 and max=1). Note: the variance of the uniform distribution with min 0 and max 1 is 1/12. Include: 1. A histogram of the results of the MC simulation 2. A density plot of a normal distribution with the appropriate mean and standard deviation 3. The mean and standard deviation of the MC simulation. ps(plz do not use chatgpt)the logit function(given as l(x)) is the log of odds function. what could be the range of logit function in the domain x=[0,1]?
- 5. The probability distribution of discrete random variable X is given by k for x = 1, 2, 3 x +1 P(X = x) %3DConsider a simple symmetric (four possible directions are equally likely) 2D random walk with the starting point of (0,0), step length=1. 1. Plot a 1000-step random walk path 2. Every time you repeat question 1, you will go through a different path, and end up with a different final location. Calculate the average distance you would cover, and the average final location you would getConsider the following procedure for initializing the parameters of a neural network: 1. Pick a random number r r = rand(1,1) * (2 + INIT_EPSILON ) – INIT_EPSILON 2. Set e =r for all i, j,l Does this work? No, because the procedure fails to break symmetry. O b. Yes, unless we are unlucky and get r = 0 (up to numerical precision). O. Yes or no, depending on the training set inputs x(i). d. Yes, because the parameters are chosen randomly.