Publications by Bonnie Cooper
Discussion Week7
DATA605: Assignment #6 Bonnie Cooper 6.1.6 A die is rolled twice. Let X denote the sum of the two numbers that turn up, and Y the difference of the numbers (specifically, the number on the first roll minus the number on the second). Show that E(XY ) = E(X)E(Y ). Are X and Y independent? library( dplyr ) #make a dataframe to hold each possible ro...
601 sym R (1408 sym/14 pcs)
Discussion Week6
DATA605: Assignment #6 Bonnie Cooper Discussion Exercise Chapter 4 #29 page 164 A student is applying to Harvard and Dartmouth. He estimates that he has a probability of .5 of being accepted at Dartmouth and .3 of being accepted at Harvard. He further estimates the probability that he will be accepted by both is .2. What is the probability that...
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Discussion Week5
Introduction to Probability Chapter 2 #5 Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density \[f(t)=\lambda e^{-\lambda t}\] where \(\lambda= 1\), so that the probability \(P(0, T )\) that a particle will appear in the next \(T\) seconds is \(P ([0, T ]) = \int _{0}^{T} \lambda e ^{−...
2942 sym
Matrix Representation
Exercise MR.C20 Compute the matrix representation of \(T\) relative to the bases \(B\) and \(C\). \[T:P_3\rightarrow \mathbb{C} ^3,\quad T(a+bx+cx^2+dx^3)= \begin{bmatrix} 2a & -3b & +4c & -2d \\ 1a & 1b & -1c & 1d \\ 3a & 0b & 2c & -3d \end{bmatrix}\] \[B = \{1,x,x^2,x^3\}\quad C= \left\{ \begin {bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin {bm...
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DATA605 HW#4
DATA605: Assignment 4 Bonnie Cooper Inverse of a Matrix / Single Value Decomposition Problem Set #1 Given a 3 × 2 matrix A: \[A = \begin{bmatrix} 1 & 2 & 3 \\ -1 & 0 & 4 \end{bmatrix}\] write code in R to compute \(\mathbf{X} = \mathbf{AA^T}\) and \(\mathbf{Y} = \mathbf{A^TA}\). Then, compute the eigenvalues and eigenvectors of \(\mathbf{X}\) ...
1292 sym R (4920 sym/50 pcs)
DATA605 HW#5
DATA605: Assignment #5 Bonnie Cooper 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. set.seed(123) B <- runif( 1 ) C <- runif( 1 ) paste( 'B = ', B ) ## [1] "B = 0.2875...
1580 sym R (2175 sym/10 pcs) 5 img
DATA605 HW#6
DATA605: Assignment #6 Bonnie Cooper (1) A box contains 54 red marbles, 9 white marbles, and 75 blue marbles. If a marble is randomly selected from the box, what is the probability that it is red or blue? Express your answer as a fraction or a decimal number rounded to four decimal places. total_marbles <- 54 + 9 + 75 P_red <- round( 54/total_ma...
4405 sym R (6717 sym/37 pcs)
DATA605 HW#7
DATA605: Assignment #6 Bonnie Cooper (1) Let \(X_1, X_2, . . . , X_n\) be \(n\) mutually independent random variables, each of which is uniformly distributed on the integers from 1 to \(k\). Let \(Y\) denote the minimum of the \(X_i\)’s. Find the distribution of \(Y\). To find the distribution, start by evaluating the probability where the min...
2258 sym R (2723 sym/11 pcs) 1 img
response p312 q3
library( ggplot2 ) res <- c( ) flips <- c( ) diffs <- c( ) for ( i in 1:1000 ){ num_of_flips <- i*10 trial <- stats::rbinom( n = num_of_flips, size = 1, prob = 0.5 ) Sn <- abs( sum( trial ) - length( trial ) ) diff <- Sn - ( num_of_flips / 2 ) percent <- round( Sn / num_of_flips * 100, 4 ) res <- c( res, percent ) diffs <-c( diffs, ...
9 sym R (432 sym/2 pcs) 2 img
Discussion Week10
DATA605: Discussion Week #10 Bonnie Cooper 11.2.14 With the situation in Exercise 13 (the homework problem), consider the strategy such that for \(i \lt 4\), Smith bets \(min(i, 4 − i)\), and for \(i \geq 4\), he bets according to the bold strategy, where \(i\) is his current fortune. Find the probability that he gets out of jail using this st...
811 sym R (1470 sym/6 pcs) 1 img