Show transcribed image text 1. (5 marks) For 2. (5 marks) In a population, 20% have blue eyes. An optician sees 225 separate patients over a Poisson random variable X with parameter Î», show that Var(X) = Î». a one week period. Use Chebyshev's inequality to find a lower bound on the probability that the number of patients with blue eyes seen over the week is between 30 and 60. State any assumptions you have made. 3. (5 marks) Suppose that X is a continuous random variable with density function given by 6(z+1 0, otherwise. (a) Show that fx satisfies the definition of a density function. (b) Determine the cuinulative distribution function of X and sketch its graph. (e) Find the expectation, the variance and the standard deviation of X 4. (5 marks) An exponential random variable x. with parameter Î». has density function: fx(r) = 0, show that E(X] and var(x) =

1. (5 marks) For 2. (5 marks) In a population, 20% have blue eyes. An optician sees 225 separate patients over a Poisson random variable X with parameter Î», show that Var(X) = Î». a one week period. Use Chebyshev's inequality to find a lower bound on the probability that the number of patients with blue eyes seen over the week is between 30 and 60. State any assumptions you have made. 3. (5 marks) Suppose that X is a continuous random variable with density function given by 6(z+1 0, otherwise. (a) Show that fx satisfies the definition of a density function. (b) Determine the cuinulative distribution function of X and sketch its graph. (e) Find the expectation, the variance and the standard deviation of X 4. (5 marks) An exponential random variable x. with parameter Î». has density function: fx(r) = 0, show that E(X] and var(x) =