In a species of animals a constant fraction of the population = 6.2 are born each breeding season and a constant fraction ẞ=4.5 die. Formulate a difference equation for the population and find out the number of individuals after fifteen seasons given that the initial number is 987. Find the closed form solution of the formulated difference equation. If the growth rate of the population is represented by r then interpret the solution obtained when i) r > 0 and ii) r < 0.

Assume that the return distribution on the two securities X and Y be as given below:

which security is more risky in the Markowitz sense. Also find the correlation coefficient of securities X and Y.

Let P=(w₁,w₂) be a portfolio of two securities. If variance of P is minimum then find the value of w₁and w₂ in the following situations.

i) ρ12 = −1

ii)σ1 = σ2

iii)ρ12 = − ,5.0 σ1 = 5.1 and σ2 = .5.2

Five securities have the following expected returns

A=20%, B=15%, C=25%, D=22%, E=18%. Calculate the expected returns for a portfolio consisting of all five securities under the following conditions

i) The portfolio weights are of equal percentage in each

ii) The portfolio weights are 32% in A and remaining are equally divided among other four securities.

Consider the following data

Use a best fit line to estimate the value of y when x = 4.5.

ompanies located on the banks of a river, dumping their chemicals waste into river, causing high levels of pollution. Local authorities passed new legislation with very high fines if the pollution in the river exceeds certain specified concentration limits. State, giving reasons, the type of modeling you will use to find a policy for discharging the waste to ensure that the concentration level never exceeds the specified limits. Also state four essentials and two non-essentials for the problem.

Prove that, for a zero-memory source with q symbols, the maximum value of the entropy is log q, which is achieved if and only if all source symbols are equiprobable. [Hint: Consider the quantity log q-H(z) and note the inequality In x ≤x-1]

Consider an 8-pixel line of intensity data, (108,139,135,244,172,173,56,99). If it is uniformly quantized with 4-bit accuracy, compute the rms error and rms signal-to- noise ratios for the quantized data.

The white bars in the test pattern shown are 7 pixels wide and 210 pixels high. The separation between bars is 17 pixels. What would this image look like after application of

i) A 3×3 arithmetic mean filter?

ii) A 7x7 arithmetic mean filter?

iii) A 9×9 arithmetic mean filter?

Consider a 3x3 spatial mask that averages the four closet neighbours of a point

(x, y), but excludes the point itself from the average.

i) Find the equivalent filter, H (u, v), in the frequency domain.

ii) Show that your result is a lowpass filter.

Prove that both 2-D continuous and discrete Fourier transforms are linear operations.

Consider the two image subsets, S₁ and S2, shown in the following figure. For V = {1}, determine whether these two subsets are (i) 4-adjacent, (ii) 8-adjacent, or

(iii) m-adjacent.

An automobile manufacturer is automating the placement of certain components on the bumpers of a limited-edition line of sports cars. The components are colour coordinated, so the robots need to know the colour of each car in order to select the appropriate bumper component. Models come in only four colours: blue, green, red, and white. Find a solution based on imaging and determine the colour of each car, keeping in mind that cost is the most important consideration.

Write an expression for 2-D continuous convolution.

Two images, f (x,y) and g (x,y), have histograms he and hg. Give the condition under which you can determine the histograms of

i) f(x,y)+g(x, y)

ii) f(x,y)-g(x,y)

iii) f(x, y)xg(x, y)

iv) f(x,y)+g(x,y)

Apply Prewitt operators and Sobel operators for the image given below:

What is Discrete Fourier Transform (DFT)? Find DFT of the function:

What do you understand by the term "Entropy" in context of any digital image? Calculate the entropy for the symbols, where probability distribution is given below:

Define the terms 'Sampling' and 'Quantization' in context of digital image processing. A medical image has size 8x8 inches, the sampling reduction is 5 cycles/mm, calculate the number of pixels required for the medical image.

Consider a linear, position-invariant image degradation system with impulse response

Supose that the input to the system is an image cosnsiting of a line of infinitesimal width located at x = a, and modeled by f(x, y) = 8(x-a), where 8 is an impulse. Assuming no noise, what is the output image g(x, y)?