The training data consists of N pairs (x1,y1),(x2,y2),··· ,(xN,yN), with xi ∈
xi will belongs to the independent variable of the training data, they are values against which value of y is measured. This are the data points which are classified based on the value that yi takes for a particular xi
The training data consists of N pairs (x1,y1),(x2,y2),··· ,(xN,yN), with xi ∈
Consider the following setting. You are provided with n training examples: (x1, y1), (x2, y2), · · · , (xn, yn), where xi is the input example, and yi is the class label (+1 or -1). However, the training data is highly imbalanced (say 90% of the examples are negative and 10% of the examples are positive) and we care more about the accuracy of positive examples. How will you modify the perceptron algorithm to solve this learning problem? Please...
Suppose that X1, X2,.... Xn and Y1, Y2,.... Yn are independent random samples from populations with the same mean μ and variances σ., and σ2, respectively. That is, x, ~N(μ, σ ) y, ~ N(μ, σ ) 2X + 3Y Show that is a consistent estimator of μ.
Consider a random sample (X1, Y1),(X2, Y2), . . . ,(Xn, Yn) where Y | X = x is modeled by a N(β0 + βx, σ2 ) distribution, where β0, β1 and σ 2 are unknown. (a) Prove that the mle of β1 is an unbiased estimator of β1. (b) Prove that the mle of β0 is an unbiased estimator of β0.
(a) Show that the points (x1, yı), (X2, y2), ..., (xn, yn) are collinear in R2 if and only if 1 X1 yi X2 Y3 rank < 2 1 Xn yn ] (b) What is the generalization of part (a) to points (x1, Yı, zı), (x2, y2, 22), ...,(Xn, Yn, zn) in R'. Explain.
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
(a) If C is the line segment connecting the point (X1,Y1) to the point (X2, y2), find the following. Jexdy-y x dy - y dx O A = (b) If the vertices of a polygon, in counterclockwise order, are (X1,Y1), (x2, y2), ..., (Xn, Yn), find the area of the polygon. O A = 3 [(x112 - – *287) + (x3X3 – x3y2) + ... + (*n – 1'n – XnYn – 1) + (xn/1 – xqYn] = {[(x112 +...
1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... < Yn be the order statistics from this sample. a) Find the joint pdf of (Y1, .. , Yn). b) Find the joint pdf of (W1, .. , Wn) where W1 = nY1, W2 = (n-1)(Y2 -Y1), W3 = (n - 2)(Y3 - Y2),..., Wn-1 = 2(Yn-1 - Yn-2), Wn = Yn - Yn-1. (c) Show that Wi's are independent and its distribution is identically...
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
(a) If C is the line segment connecting the point (X1,Y1) to the point (X2, y2), find the following. e x dyr dy - y dx xly2 - x2y1 x A= A= (b) If the vertices of a polygon, in counterclockwise order, are (X1,Y1). (X2, y2), ..., (X, Yn), find the area of the polygon. [0x271 – 1/2) + (x392 – x2Y3) + .. + ... + (xnxn-1 - xn-1n) + (*11n – Xnxx)] + x2+1) + (x2y + x372)...
Let the independent normal random variables Y1,Y2, . . . ,Yn have the respective distributions N(μ, γ 2x2i ), i = 1, 2, . . . , n, where x1, x2, . . . , xn are known but not all the same and no one of which is equal to zero. Find the maximum likelihood estimators for μ and γ 2.