Let Xn, Yn be independent simple random walks. Let Zn be (Xn, Yn) truncated to lie in the region Xn ≥ 0, Yn ≥ 0, Xn + Yn a where a is integral. Find the stationary distribution of Zn.
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- Let X be a random variable with uniform distribution on the interval [-2,2]. Let Y be defined as Y = X5. Calculate the pdf of Y.Let random variable X be uniform in the interval (0, 1). Define random variable Y = aX + b where a not 0.Let X ~ U[0,1] and Y = -βln(1-X). What is the distribution of Y? Justify.
- Let Y1 < Y2 < Y3 be the order statistics of a random sample of size 3 froma distribution having the pdf f(x) = 2x, 0 < x < 1, zero elsewhere. Show thatZ1 = Y1/Y2, Z2 = Y2/Y3, and Z3 = Y3 are mutually independent.Let the random variable X fit uniformly in the interval (-1,1). Find the distribution of the random variable X².Let X be a (continuous) uniform random variable on the interval [0,1] and Y be an exponential random variable with parameter lambda. Let X and Y be independent. What is the PDF of Z = X + Y.
- Assume that X and Y are independent random variables where X has a pdf given by fx(x) = 2aI(0,1)(x) and Y has a pdf given by fy(y) = 2(1– y)I(0,1)(y). Find the distribution of X + Y.Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of Y
- Let X and Y be independent v.a with uniform distribution on the interval (0, 20) and (0, 30) respectively. Find the function f_X+Y (z).Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; isb) Let Z₁-N(0,1), and W₁ = Y~N(0,1), for i=1,2,3,...,10, then: dx dy i) State, with parameter(s), the probability distribution of the statistic, T = - 154 ii) Find the mean and variance of the statistic T = ₁² 10 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ß such that P(T> B) = 0.01, where T = ₁2₁² +².