KIndly help me solve the following questions below
1.Question:
The outlet stream of a continuous chemical reactor is sampled every thirty minutes and titrated. Extensive records of normal operation show the concentration of component A in this stream is approximately normally distributed with mean 41.2 g/L and standard deviation 0.90 g/L.??
??a) What is the probability that the concentration of component A in this stream will be more than 42.3 g/L????
??b) Five determinations of concentration of component A are made. If the mean of these five concentrations is more than 42.3 g/L, action is taken. What is the level of significance associated with this test????
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c) State the null hypothesis and the alternative hypothesis that fit the test described in part (b).??
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d) The test in part (b) is applied. Now suppose the true mean has changed to 43.5 g/L with no change in standard deviation. What is the probability of a Type II error???
5. [12 marks] Consider the Markov chain with the following transition matrix. 0 0.5 0.5 0.5 0 0.5 0.5 0.5 0 (a) Draw the transition diagram of the Markov chain. (b) Is the Markov chain ergodic? Give a reason for your answer, (c) Compute the two step transition matrix of the Markov chain. (d) What is the state distribution 72 for t = 2 if the initial state distribution for t = 0 is no = (0.1, 0.5, 0.4) ?? (e) What is the average time /1,1 for the chain to return to state 1? Remember: Mij = 1 +) Pikikj .Consider the 3-state Markov chain described by the graph below, where pe [0, 1]. 1 p 3 2 Obtain a full classification of the states in this Markov chain (classes, recurrence, tran- sience, periodicity) and when appropriate obtain the steady-state probabilities.4.1 Two Types of Random Variables 4.2. Probability Distributions for Discrete Random Variables 1. What is a random variable? 2. Give examples for discrete random variable, continuous random variableProblem 2: 2.1) Write down the mathematical definition of each of the following. (Be precise in your definitions. Each definition should be a mathematical statement.) (a) A random variable (b) The pdf of a random variable. (Specify whether the random variable is discrete or continuous.) (c) The pmf of a random variable. (Specify whether the random variable is discrete or continuous.) (d) The cdf of a random variable. (Specify whether there are any restrictions on whether the random variable is discrete or continuous.) (e) the expected value of a random variable. (Give separate expressions for discrete and continuous random variables.) (f) the expected value of a function of a random variable. (Give separate expressions for discrete and continuous random variables.) 2.2) Write down the mathematical definition of each of the following random variables. (Each definition should be a mathematical statement.) For any parameters of the distribution, explain the meaning of each parameter. Specify whether each of these is a discrete or continuous random variable. (a) a Bernoulli random variable (b) a Binomial random variable (c) a geometric random variable (d) a Poisson random variable (e) a uniform random variable (f) an exponential random variable (g) a normal (or Gaussian) random variableDetermine whether the following value is a continuous random variable, discrete random variable, or not a random variable. a. Is the amount of rainfall in a month is a discrete random variable, continuous random variable, or not a random variable? O A. It is a discrete random variable. O B. It is a continuous random variable. O C. It is not a random variable. b. Is the list of books listed for a book club event a discrete random variable, continuous random variable, or not a random variable? O A. It is a discrete random variable. O B. It is a continuous random variable. O C. It is not a random variable. c. Is the number of cars a dealership owns a discrete random variable, continuous random variable, or not a random variable? O A. It is a discrete random variable. O B. It is a continous random variable. O C. It is not a random variable
