We have to maximize subject to . The Lagrangian is
Taking partial derivatives with respect to and equating to 0,
Thus the maximum entropy occurs, when the distribution is uniform
The maximum entropy is
3. Let X be the random variable characterized by a probability distribu tion p (P, ...p)...
upposes is a binomial random variable with n = 5 and , 2,3,4, and , using the foll. Compute p(x) for x 0, 1, 2, 3, 4, and 5, using a. List th S for Success and F for Failure on each trial) corresponding to each value of x, assign probabilities to each sample point, and obtain p= wing two methods e sample points (take p(x) by adding sample-point probabilities tion to obtain p(x) b. Use the formula for the...
In information theory Shannon entropy is defined by H(x) = -Sum(P(x)*log(P(x)) where P is probability mass function of random variable x, and log to base 2. Given loaded die with P(6)=0.5 and P(1)=P(2)=P(3)=P(4)=P(5), compute entropy of observed rolls 654266. Note: To answer the question you do not need to know any more information about Shannon entropy.
. The central limit theorem provides us with a tool to approximate the probability distribu- tion of the average and the sum of independent identically distributed random variables In the following questions, use the "is approximately" sign ะ when you apply the central limit theorem. Use Table 4 on p.848 of Wackerly to determine probabilities. (a) Let X1, , X500 be independent Gamma(α-1, B-2) distributed random variables. Σίκο Xi. After simulating 500 values, we find a realization of X500 of...
2. Let X be a Bernoulli random variable with probability of X -1 being a. a) Write down the probability mass function p(X) of X in terms of a. Mark the range of a (b) Find the mean value mx(a) EX] of X, as a function of a (c) Find the variance σ剤a) IX-mx)2) of X, as a function of a. (d) Consider another random variable Y as a function of X: Y = g(X) =-log p(X) where the binary...
1) Let X be a random variable that assume the values {0,1,2,3} with probability pi=1/14i^2. Determine in simplify fraction 2) Let X be a random variable that assume the values {-1,0,1} with probability p-1=1/4, p0=1/2, and p1=1/4 Determine P(X</-1|X</0)= Sea X una variable aleatoria que asume los valores {0,1,2,3} con probabilidad pe= -12. Determina en fracción 14 simplificada: - ש o=y 1 Sea X una variable aleatoria que asume los valores {-1,0,1} con probabilidad p-1 = 7: Po = ŻY...
Let X be a random variable that takes values x = (xi,... , xn) with respec- tive probabilities p (pi,. .. , pn). Write two R functions mymean (x,p) and myvariance (x, p), which find the mean and variance of X, respec- tively. Use your function to find the mean and variance of the point value of a random Scrabble tile, as in Example 4.1 Let X be a random variable that takes values x = (xi,... , xn) with...
Compute the probability for a random variable X with µ=10 and σ=2. Calculate P(9 < X < 11).
Let descrete random variable X ~ Poisson(7). Find: 1) Probability P(X = 8) 2) Probability P(X = 3) 3) Probability P(X<4) 4) Probability P(X> 7) 5) ux 6) 0x Show your explanations. Displaying only the final answer is not enough to get credit. Note: round calculated numerical values to the fourth decimal place where applicable.
(Use computer) Let X represent a binomial random variable with n = 110 and p = 0.19. Find the following probabilities. (Round your final answers to 4 decimal places.) a. P(X ≤ 20) b. P(X = 10) c. P(X > 30) d. P(X ≥ 25) (Use Computer) Let X represent a binomial random variable with n = 190 and p = 0.78. Find the following probabilities. (Round your final answers to 4 decimal places.) Probability a....
Problem 3. Let X be a discrete random variable, with probability distribution P(X X1) = 0.95, P(X X2) = 0.05. Determine x, and X2 such that E(X-0 and σ2(X) = 7.