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T_a = α T_o + (1-α) T_b,

where α is a scalar parameter.

a) Find the value of α which gives the minimum variance estimate. Verify your answer by showing that the variance of the analysis error σ^2_a is given by

σ^2_b = (σ^2_b * σ^2_o - ρ^2) / (σ^2_b + σ^2_o - 2ρ)

b) Show that if the background and observation errors are uncorrelated (ρ= 0) then the analysis error T_a - T_b is uncorrelated with the difference between the observation and the background T_o - T_b