(a) Assume that the distribution of female height is N(μF, σ 2,F) and the distribution of male height is N(μM, σ 2,M). In addition, assume that the proportions of female and male are the same. (i) (3 marks) Find the CDF of human height. (ii) (2 marks) Find the PDF of human height. (iii) (5 marks) Assume that μF = 165 cm, μM = 175 cm, and σF = σM = σ for some constant σ > 0. Generate a plot containing the plots of the PDF of human height for the following σ values: 4, 5, 6. Now do the same for these σ values: 4.5, 5, 5.5. Describe how the number of modes of the PDF changes as σ changes. (iv) (5 marks) Based on your investigation above, formulate a condition of when the PDF of human height is bimodal. State the condition as an inequality involving μF, μM and σ. Generate two additional plots to support your conjecture by choosing different μF,μ M and σ values.