Publications by Vinayak Kamath
Data605-Week2-Discussion2-kamath
Exercise- Determinants - C25 #defining the matrix: A <- matrix(c(3, 2, 2, -1, 5, 0, 4, 1, 6), 3, 3) A ## [,1] [,2] [,3] ## [1,] 3 -1 4 ## [2,] 2 5 1 ## [3,] 2 0 6 #Doing manually : (3 *(5*6 - 1*0) ) - ( -1 *(2*6 - 2*1) ) + ( 4 *(2*0 - 2*5)) ## [1] 60 #Using R funtion to validate the output: det(A) ## [1] 60 ...
39 sym R (312 sym/6 pcs) 1 img
Data605-Week1-Discussion1-kamath
Exercise- Matrix - C10 A + B A+B ## [,1] [,2] [,3] ## [1,] 4 6 -2 ## [2,] 4 -3 5 A + C #A+C A and C are not the same size.A+C is not defined. B^t + C B1 + C ## [,1] [,2] ## [1,] 5 2 ## [2,] 6 -6 ## [3,] -1 7 A + B^t #A + B1 A and B^t are not the same size. A + B^t is not defined. (BetA)C Beta * C #...
235 sym R (588 sym/16 pcs) 1 img
Data605-Week1-HomeWork1-kamath
Problem set 1 Calculate the dot product u:v where u = [0.5; 0.5] and v = [3;-4] u <- matrix(c(0.5, 0.5), 1, 2) v <- matrix(c(3, -4), 1, 2) #print("dot product u:v is ") (0.5 * 3) + (0.5 * -4) ## [1] -0.5 What are the lengths of u and v? Please note that the mathematical notion of the length of a vector is not the same as a computer science ...
1317 sym R (1110 sym/12 pcs) 1 img
Data605-Week8-Discussion8-Kamath
Exercise- LAW OF LARGE NUMBERS - 1 A fair coin is tossed 100 times. The expected number of heads is 50, and the standard deviation for the number of heads is (100 · 1/2 · 1/2)1/2 = 5. What does Chebyshev’s Inequality tell you about the probability that the number of heads that turn up deviates from the expected number 50 by three or more sta...
597 sym R (232 sym/2 pcs)
Data605-Week6-Discussion6-Kamath
Exercise- CONDITIONAL PROBABILITY - #2 Solution: A Coin can land as Head (H) or Tail (T); two choices. With three tosses, we can have belwo combinations: 2 x 2 x 2 = 8 combinations as below 1. H H H 2. H H T 3. H T H 4. H T T 5. T H H 6. T H T 7. T T H 8. T T T We can now solve the below as: What is the pr...
60 sym R (1218 sym/1 pcs) 1 img
Data605-Week5-Discussion5-Kamath
Exercise- Discrete Probability Distributions - 1.2 10 Solution: Let P(S) denote the probability that a bill passes the Senate; let P(H) denote the probability that a bill passes the House of Representatives. we have as below: P(H \(\cup\) S) = P(H) +P(S) − P(H \(\cap\) S) Solving for P(H\(\cap\)S) yields P(H \(\cap\) S) = P(H) + P(S) − P(H \...
582 sym 1 img
Data605-Week4-HomeWork4-kamath
Problem set 1 . #defining the matrix: A <- matrix(c(1, -1, 2, 0, 3, 4), 2, 3) A ## [,1] [,2] [,3] ## [1,] 1 2 3 ## [2,] -1 0 4 # Compute X and Y using built-in commands X <- A%*%t(A) X ## [,1] [,2] ## [1,] 14 11 ## [2,] 11 17 Y <- t(A)%*%A Y ## [,1] [,2] [,3] ## [1,] 2 2 -1 ## [2,] 2...
345 sym R (4499 sym/35 pcs) 2 img
Data605-Week4-Discussion4-Kamath
Exercise- Representations- C10 #defining the matrix B and v: B <- matrix(c(2, -2, 2, 1, 3, 1, 3, 5, 2), 3, 3) B ## [,1] [,2] [,3] ## [1,] 2 1 3 ## [2,] -2 3 5 ## [3,] 2 1 2 v <- matrix(c(11, 5, 8), 3, 1) v ## [,1] ## [1,] 11 ## [2,] 5 ## [3,] 8 # With augmented matrix we have as below: A <- m...
41 sym R (718 sym/10 pcs) 1 img
Data605-Week3-HomeWork3-kamath
Problem set 1 . #defining the sample matrix: A <- matrix(c(1, -1, 0, 5, 2, 0, 1, 4, 3 ,1, -2, -2, 4, 3, 1, -3), 4, 4) A ## [,1] [,2] [,3] [,4] ## [1,] 1 2 3 4 ## [2,] -1 0 1 3 ## [3,] 0 1 -2 1 ## [4,] 5 4 -2 -3 #run the function qr() qr(A)$rank ## [1] 4 #Alternative: load the Matrix packa...
172 sym R (1186 sym/20 pcs) 5 img
Data605-Week5-HomeWork5-kamath
Problem set 1 Choose independently two numbers B and C at random from the interval [0, 1] with uniform density. Prove that B and C are proper probability distributions. Note that the point (B,C) is then chosen at random in the unit square. Find the probability that B + C < 1/2 B + C = 1 passes through the points (1/2, 0) and (0, 1/2). ==> B + C ...
1295 sym