COMPS-(a) Given two sorted arrays arr1, arr2, with

python

Explanation

The provided code defines a function to merge two sorted arrays into one sorted array and includes a main function to test the merging function with two use cases.

Step-by-step Instruction

▪

Initialize a result array res and two pointers i and j to keep track of the current index in each input array

▪

Use a while loop to iterate through both arrays until all elements are processed

▪

Inside the loop, use conditional statements to decide which element to add to res based on the comparison of elements pointed by i and j

▪

Increment the pointers i or j after adding the respective element to res

▪

After the loop, return the merged and sorted array res

▪

Define the main function main to test the merge_sorted_arrays function with specific use cases

▪

Use assert statements to verify that the function returns the correct merged and sorted array for the given use cases

▪

Call the main function to execute the tests

Time Complexity

The time complexity of the merge_sorted_arrays function is O(n1 + n2), where n1 and n2 are the lengths of the input arrays, because it processes each element of both arrays exactly once.

Space Complexity

The space complexity is O(n1 + n2) because it creates a new array res that contains all elements from both input arrays.

To address the optimization problem presented in the images, we need to find a specific value of

k

that minimizes the function

f

. The function

f

is defined as

f(k, b_1, \ldots, b_k, d_1, \ldots, d_k, c_1, \ldots, c_k) = \exp(\|A - B\|_F^2) + k(m + n + 1)

, where

\|.\|_F

denotes the Frobenius norm,

A

is a given matrix of size

m \times n

, and

B

is a matrix constructed from the outer products of vectors

b_i

,

d_i

, and

c_j

.

The function

g

is defined as

g(k, b_1, \ldots, b_k, d_1, \ldots, d_k, c_1, \ldots, c_k) = \exp(\|A - B\|_F^2)

. Minimizing

g

is equivalent to minimizing

\|A - B\|_F^2

, which, according to the Eckart-Young-Mirsky theorem, is achieved by the truncated SVD of

A

. The minimum of

g

occurs when

k = n

, as

g

is a decreasing function with respect to

k

.

The second part of the function,

k(m + n + 1)

, is an increasing function with respect to

k

, and its minimum occurs when

k = 1

.

The optimization problem requires finding a trade-off between minimizing

g

and minimizing

k(m + n + 1)

. This trade-off is represented by the objective function

\text{argmin}_k \{ \exp(\|A - B\|_F^2) + k(m + n + 1) \}

.

To solve this optimization problem in R, one would typically use optimization functions such as "optim" or "optimize", which can handle the trade-off between the two parts of the function

f

to find the optimal value of

k

.

you need to write R code to solve this question

r

Explanation

The R code provided defines a function "merge_sorted_arrays" that takes two sorted arrays as input and returns a single sorted array containing all elements from both arrays. The main function "test_merge_sorted_arrays" tests the "merge_sorted_arrays" function with two use cases to ensure it works correctly.

Step-by-step Instruction

▪

Initialize an empty array to hold the merged result and set pointers to track the current position in each input array

▪

Use a while loop to iterate through both arrays, comparing and appending the smaller element to the merged array

▪

After one array is exhausted, append the remaining elements of the other array to the merged array

▪

Define the `test_merge_sorted_arrays` function to test the `merge_sorted_arrays` function with predefined use cases

▪

Call the test function to execute the tests and validate the functionality of the merge function

Time Complexity

The time complexity of the merge function is O(n + m), where n and m are the lengths of the two input arrays, because each element from both arrays is visited at most once.

Space Complexity

The space complexity is O(n + m) because a new array is created to store the merged result of the two input arrays.

A

Key Concept

Mixed Distribution

Explanation

The joint distribution of X and Y is not continuous because X is a discrete random variable and Y is a continuous random variable. When combined, they form a mixed distribution.

B

Key Concept

Joint Distribution Function for Mixed Random Variables

Explanation

To find the joint distribution function

F_{XY}(x, y)

, we need to consider the distribution of X and Y separately due to their independence, and then combine them. For x < 0,

F_{XY}(x, y) = 0

. For

x \geq 1

, F_{XY}(x, y) = P(Y < y). For 0 \leq x < 1, F_{XY}(x, y) = p + (1 - p)P(Y < y). Since Y has an exponential distribution, P(Y < y) = 1 - e^{-\lambda y} for

y \geq 0

. Thus,

F_{XY}(x, y)

can be defined piecewise based on the value of x.

Here is the code to calculate

F_{XY}(x, y)

for given values of

x

,

y

,

p

, and

\lambda

:

python

for this question, please use latex mathematics equations to prove it

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