Q 1.3.3

Solution

a)

  • Comments: Perfect fit for 7 points

b)

  • Comments: Perfect fit for 7 points

c)

  • Comments: Perfect fit for 3 points

d)

  • Comments: not fit for 7 points

Q 1.4.3

Solution

a)

  • linearized approximation of P function

b)

c)

d)

  • Assume \(c<0\)

Q 1.5.2

Solution

a)

Three linear functions :

  • \(\mathrm{T} = \beta_{0}+ \beta_{1}(t-2007)\)

  • \(\mathrm{T} = \beta_{0}+ \beta_{1}(t)\)

  • \(\log(\mathrm{T}) = \beta_{0}+ \beta_{1}\log(t)\)

b)

Case-1

OLS linear Regression fitting :

  • \(a= 383.52\)
  • \(b= 1.9552\)

Case-2

OLS linear Regression fitting :

  • \(c= 0.46475\)
  • \(d= \pm 0.0623\)
  • \(e= 1/0.014768 = 67.71398\)

c)

d)

Therefore, by increasing 10% of \(CO_2\) concentration of ppmv , the Global Temperature Anomaly T will increase from \(0.4^oC\) to \(0.7584^oC\). Hence the net increment of T is \(0.3584^oC\).

Q 3.2.3

Solution

a)

b)

Q 3.4.4

Solution

a)

b)

Q 4.3.12

Solution

a)

b)

c)

  • Gamma Function:

  • Hence the pdf of \(p(x)\) density distribution can be expressed with with Gamma Function.

d)

e)

  • The probability from 0 to \(3a_0\) found to be \(0.93803\)

Q 4.5.3

Solution

a)

  • I guess “Filtered pdf function” from this text book means smoothing the pdf function. I found this text book is too outdated and make readers too confused. A lots of statistical concepts and terminology wrongly used in this outdated text book !!!

  • Larger the filtered interval for pdf function, greater the smoothing effect(averaging) for the pdf, but loosing the precision of the pdf function.

  • Three Gaussian filter interval:

    • \(\delta x = 2\)
    • \(\delta x = 5\)
    • \(\delta x = 30\)

Q 5.3.4

Solution

a)

  • the y0 = 1, y1= 1, y2 = 2, y3 = 3, y4 = 5 , y5 = 8
  • total no of pairs for first five months + initial y0 = 1 +1+ 2 +3 +5 + 8 = 20 pairs

b)

  • the n-month Fibonacci Paired Rabits:
  • \(F_0 = F_1 = 1\)
  • \(F_n := F_{n-1} + F_{n-2}\)

c)

d)

\[\begin{align*} F_{12} &= \left(\frac{5 +\sqrt{5}}{10} \right)\left(\frac{1 +\sqrt{5}}{2} \right)^{12}+\left(\frac{5 -\sqrt{5}}{10} \right)\left(\frac{1 -\sqrt{5}}{2} \right) ^{12} \\ F_{12} &=232.9991 -0.0008583723 = 232.9983 \approx 233 \ \end{align*}\]

  • The rabbit pairs after one year found to be 233.

Q 5.4.2

Solution

a-and-b)

  • non zero stable equilibrium point occurred at \(y = \frac{log(a)}{\beta}\) for \(a> 1\).
  • the equilibrium solution is stable provided that the condition of \(1< a < e^{2}\) is hold.