(4) If X and Y are two random variables, then the joint moments can be derived from the joint moment generating functions as mnk = ank Mxy (V₁, V₂)bne Yzeldshav mobustown?? XY = 0(₂) k @v,"av "
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- Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?Let X1, X2, ... Xn random variables be independent random variables with a Poisson distribution whose parameters are l1, l2, ... ln, respectively. Which of the following is the moment generating function of the random variable Z defined as (the little image)?If X, Y are normally distributed independent random variables, then show that W = 2X - Y is normally distributed. Do not use moment generating function, only use convolution formula.
- A single observation of a random variable having a hypergeometric distribution with N = 7 and n = 2 is used to test the null hypothesis k = 2 against the alternative hypothesis k = 4. If the null hypothesis is rejected if and only if the value of the random variable is 2, find the power of the test.Let X and Y be independent Poisson random variables with means 2 and 8, respectively. More explicitly, their probability mass functions are given by Px (k)=e¹ PY (k)= e and let Z=X+Y Find the variance of X given that Z=10, that is Var(X|Z=10). Express the result with two decimal points, Ex 12.34 Yaret: for k=0,1,... for k=0, 1,...Show that the random process X(t) =cos(2π fot + θ) Where θ is an random variable uniformly distributed in the range {0, π/2, π, π/3} is a wide sense stationary process .
- A park ranger is searching for bears in a region of the park where on average there are 5 bears per square mile. The bears are solitary independent creatures, so it is reasonable to assume that the numbers of bears in disjoint regions are independent unknowns and that the number expected in any region is proportional to the area of the region. The ranger can also assume that in a very tiny region, say a square inch, it is impossible to find more than one bear. What is the probability (two decimal place accuracy) that he finds less than 12 bears in a given 2 square mile region of the park?a data managment company records a large amount of data. Historically, there are 1.5 errors per 100 pages of data. Let x be the random variable representing the number of errors in 100 pages. i) if X has a poisson distributiuon, write down value of its paramaters ii) what is the probability that there are three errors in a randomly selected 100 pages? ii) what is the probability that there are three or more errors in a randomly selected 200 pages?According to the Maxwell–Boltzmann law of theoret-ical physics, the probability density of V, the velocity of a gas molecule, isf(v) =⎧⎨⎩kv2e−βv2for v > 00 elsewhere where β depends on its mass and the absolute tem-perature and k is an appropriate constant. Show that the kinetic energy E = 1 2mV2, where m the massof the molecule is a random variable having a gammadistribution.