Question

To prevent weeds in the grain field, a farmer has added a certain amount of θ (g / l) herbicide to the water he injects over the field. He thinks the height of the grains is a bit lower than usual and wants to adjust the amount of herbicide to next year, but he can't remember how much he has used. He asks you for help finding out how much he has spent. According to the manufacturer of the product, it is stated that the height of each grain straw y follows the distribution
f (y; θ) = (θ/(2*sqrt(y)))* exp (-θ*sqrt(y)/2), θ> 0, y> 0,
where gen is a mild herbicide added to the water. You pick out 25 random straws from the field and measure the height of them. Using the probability maximization method, you calculate an estimate of the amount of herbicide he has used

Answer 1

solution ,

height of each grain straw y follows distribution

f(y;theta)=[theta/2sqrt(y)]*e^-theta sqrt(y)/2 where theta>0 and y>0

and Amount of herbicide=theta(g/l)

if we pick 25 and measure height of them then we are to calculate an estimate of the amount of herbicide he has used by using probability maximizing method as follows

the pdf for normal distribution is

f(x)=1/sqrt(2pisigma)*e^-1/2(x-mu/sigma)^2

so on comparing this to above we find estimate mean =E(X)=mu=theta

and var( sigma)^2=theta/y thannks