Given the random variable x- f(x)=ae for x20. Estimate the moment generating function of x.
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- Find the average rates of change of f(x)=x2+2x (a) from x1=3 to x2=2 and (b) from x1=2 to x2=0.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 Y > 0 be a continuous random variable representing time from regimen start to bone-marrow transplant. Everyone does not survive long enough to get the transplant. Let X > 0 be a continuous random variable representing time from regimen start to death. We can assume X ⊥ Y and model time to death as X ∼ Exp(rate = θ) and time to transplant as Y ∼ Exp(rate = µ). Where Exp(rate = λ) denotes the exponential distribution with density f(z | λ) = λe−λz for z > 0 and 0 elsewhere - with λ > 0. Find the probability that the patient would die before receiving transplant.
- Suppose that the random change in value of a financial asset is X over the first day and Y over the second. Suppose also that Var(X) =18 and Var(Y) = 26 In this case, the total change in the value over these two days is given by X +Y. Do you have enough information to compute Var(X +Y)? If so, compute this value. If not, explain what additional information you need to do so.Y, = XB + ɛ, Show that the model variance in model Yi unbiased estimator ofSuppose that f(x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.
- Use the geometric distribution to derive E(Y), E(Y^2), and V(Y) from the Poisson distributionlet x be a random variable with moment generating function Mx(t)=(0.6 + 0.4e^t)^20 then the variance of x isLet the probability function of the random variable X be defined as f(x)=cx^4 for x > 1, otherwise f(x)=0. Calculate the constant c, the expected value and the variance.
- Let X₁, X₂ be IID with \Exp(1), the standard exponential distribution. Show that Z = X₁/X₂ has an F-distribution. Determine the degrees of freedom of this F- distribution.The pdf is given as follows: f(x) = { (1/6)e^(1/6x) ; x > 0 0; ew] (i) Find the mean and variance of X using the pdf. (ii) Find the moment generating function of X. (iii) Find the mean and variance of X using the moment generating function.Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.
