Let Y1 x1 Yn xn be the data where Yi are independently an

Let (Y_1, x_1),. . . ., (Y_n, x_n) be the data where Y_i are independently and exponentially distributed random variables in the following way: Y_i Ex(lambda x_i), i = 1, 2,. . . ,n f_Y_i = lambda x_i e^-lambda x, y 1y > 0. The known constants (x_i) are strictly positive. I have derived the maximum likelihood estimalot for lambda. I think it is Exponential distributions arc often used to model survival times. A researcher wants to find the relationship between the price of a light bulb and its life span. She samples 40 different lieht bulbs and hypothesizes the above model, whereby for i = 1,. . . .,40 x_i = 1/price of lightbulb i Y_i = life span of light bulb i (in years) The plot of the relationship between price and lifespan is given. Figure 1: 40 measurements of light bulb life spans versus their price. Given that Sigma i = 1 40 x_iy_i = 80, calculate the numeric 95% confidence interval for lambda based on asymptotic distribution of the likelihood ratio statistic. Draw this confidence interval in a plot of the likelihood ratio statistic versus lambda. You are allowed to \"read off\" the numeric CI from this plot, but provide the inequality that you would like to solve.

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