(Solved): 1. (20 marks, Sec. 3.4) Learning Model In a learning model, two responses \( A \) and \( B \) are p ...

1. (20 marks, Sec. 3.4) Learning Model In a learning model, two responses \( A \) and \( B \) are possible for each of a series of observations. If there is a probability \( p \) of getting response \( A \) in any individual observation, the probability of getting response \( A \) exactly \( n \) times in a series of \( m \) observations is \[ F(p)=p^{n}(1-p)^{m-n}, \quad 0 \leq p \leq 1 . \] The maximum likelihood estimate is the value of \( p \) that maximizes \( F(p) \). (a) What is the maximum likelihood estimate? 2. (20 marks, Sec, 3.4) Speed of Flight The power \( P \) required by a bird to maintain flight is given by the model 1 \[ P(v)=\frac{w^{2}}{2 \rho S v}+\frac{1}{2} \rho A v^{3}, \] where \( v \) is the relative speed of the bird, \( w \) is the weight of the bird, \( \rho \) is the density of the air, and \( S \) and \( A \) are positive constants associated with the bird's size and shape. (a) What relative speed will minimize the power required by the bird? (b) Your solution should confirm that relative speed \( v \) that minimizes power would be - larger if \( w \) increases (direct variation), and - smaller if \( \rho \) increases (inverse variation). What variations (be specific with the power of \( w \) and \( \rho \) ) do you find in your answer? Are the above relationships confirmed?