4. Random Process Let X (1) be a Random process, with mean x (t) = (t-1) and autocovariance Rx (11, 12) at time steps t₁, t₂ as follows: Rx (t₁,1₂)=t₁ + t₂ Find a. Mean of X(t) at time step 2. b. Autocovariance at time steps (1,5), variance of X(t) at time step 2 c. P(X(2) ≤ 1)
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- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; isIf X is a Poisson variable such that P(X =2) = 9P(X= 4) + 90P(X = 6), find the mean and variance of X.
- If X is a random variable with pdf f(x) = 2x − 2 where x = (1, 2), find the variance of Y = 2X - 3.Suppose that f(x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.Find the minimum mean square error forecast Y(1), forecast error e, (1) and Varfe, (1)1 for the following modes. Y, = 0.8Y, +e,. Y, = 3+21+e,.
- Suppose that f (x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.An investor has found that company1 have an expected return on E(X) = 4% and variance for the return equal V(X) = 0.49. Company 2 has E(Y) = 6% and variance V(Y) = 0.64. The correlation between the companies return is ρ(X,Y) = 0.3. The investor wants to invest p (0<p<1) in company1 and (1-p) in company2. The combined investment have a return: R = pX + (1-p)Y. Let p=0.4 such that R= 0.4X +0.6Y. Find the Expectation and variance of R. how are these results in comparison with X and Y separatley?The random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−34. Calculate Var(X+Y) with a full explanation
- If a random variable X has the moment generating function Mx (t)= 2 - ť Determine the variance of X.3. Let Y be the number of speeding tickets a YSU student got last year. Suppose Y has probabilitymass function (PMF)y 0 1 2 3fY (y) 0.12 0.13 0.33 0.42(a) What is the probability a YSU student got exactly one ticket?(b) What is the probability a YSU student got at least one ticket?(c) Compute µY , the mean of Y .(d) Find the variance and standard deviation of Y .(e) What is the probability that Y exceeds its mean value?Show that variance o? = (x²) – ((x))²