Assume that N belongs to the (a,b,0)-class. Let Nm be the ZM random variable of N with po and p = cpk, k = 1,2,.... Show that c= 1-pm 1-po
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- M112. a) Let X and Y be independent identically distributed binomial random variables, X~ Bin(3, 0.4) and Y ~ Bin(3, 0.4). i) Compute the probability P(X = 0). ii) Compute the probability that the product XY equals 0, that is, compute P(XY = 0). b) The actual amount of jam (in g) that a filling machine puts into "150 g" jars may be looked upon as a random variable having a normal distribu- tion with σ = 6 and mean μ = 155. What proportion of jars, in the long-run, contain less than 150 g? N(0, 1), Give your answer in terms of the function, where for Z (z) = P(Z), and a give a sketch of the normal curve with a shaded area that corresponds to the required probability.A QR code photographed in poor lighting, so that it can be difficult to distinguish black and white pixels. The gray color (X) in each pixel is therefore coded on a scale from 0 (white) to 100 (black). The true pixel value (without shadow) the code is Y = 0 for white, and Y = 1 for black. We treat X and Y as random variables. For the highlighted pixel in the figure is the gray color X = 20 and the true pixel value is white, i.e. Y = 0. We assume that QR codes are designed so that, on average, there are as many white as black pixels, which means that pY (0) = pY (1) = 1/2. In this situation, X is continuously distributed (0 ≤ X ≤ 100) and Y is discretely distributed, but we can still think about the simultaneous distribution of X and Y. We start by defining the conditional density of X, given the value of Y : fX|Y(x|0) = "Pixel is really white" fX|Y(x|1) =" Pixel is really balck " Use Bayes formula as given in the picture and find the probability for x = 20 like in the picture.
- Suppose that X;; i = 1,2, 3, .. n are independent random variables with a common uniform distribution over (0, 1) and Yg is distributed over (0, Yg -1) with Y, =1,k = 1,2, ..., n. Let Z = k = 1,2,3, .n R.D.R EGALA STAT 203 March 4, 2021 Show by mathematical induction induction or any other method that Zg has probability density function (-In z)*-1 fR(z) = k = 1,2,3, . n (k – 1)! Hence deduce that Zg and Y, have the same distribution (k = 1,2,3, ..n)Let X1,..., Xn be iid random variables from N (0, o²) with pdf ,2 1 exp 2TO2 f(x; o²) = -00 0. 202 Use the factorization theorem to show that X? is a sufficient statistic for o?. i=15
- Let X₁, X2,..., Xn be a random sample of size n from Beta (0, 1) 12 Show that the random variable - 20 Inx, has a chi-square distribution with 2n defrees of freedomLet U1 and U2 be two iid U(0,1) random variables. Show that U1 and 1 - U2 have the same distribution by showing their CDFs are the sameSuppose X is a discrete random variable. Let the pmf of X be equal to 5 - x f(x) = x = 1,2,3,4 10 Find the cdf of X, that is F(X).
- Let Y1, Y2, Y3 denote independent exponential random variables with respective rates A; for i e {1,2, 3}. Answer the following questions; (a) Compute E[Y1 +Y2 +Y3]Y1 > 1, Y2 > 2, Y3 > 3]. Your answer should be an expression using A;'s. (b) Compute the probability that Yı is the smallest among three random variables. Hint: P(X < Y) = S" Px,y (x, y)dædy where Px,Y (x, y) is the joint density function.Suppose that X is a random variable * ? having the following m.g.f., Find Var(x) 7t_e4t M, (t) = eT 173 O 5 O 413 O 3148 O 55 OLet X1, X2,....X, be a simple random sample from N(H,02), where o unknown, then the X is the MVUE of u Select one: True False
