Given a coin with the probability p of landing heads, where p is unknown and we need to estimate its value through data. In our data collection model, we have n independent tosses, result of each toss is either Head or Tail. Let X denote the number of heads in the total n tosses. Now we conduct experiments to collect data and find out that X = k. Then we need to find . the estimation of p. (a) Assume p is an unknown constant. Find through the MLE (Maximum Like- lihood Estimation) rule. (b) Assume Р is a random variable with a prior distribution Beta(a, b), where a and b are known constants. Find p through the MAP (Maximum a Posterior Probability) rule (c) Assume p is a random variable with a prior distribution Beta(a, b), where a and b are known constants. Find p through the MMSE (Minimum Mean Square Estimate) rule. 2 19 H SR #

How hot is the air in the top (crown) of a hot air balloon? Information from Ballooning: The Complete Guide to Riding the Winds, by Wirth and Young (Random House), claims that the air in the crown should be an average of 100°C for a balloon to be in a state of equilibrium. However, the temperature does not need to be exactly 100°C. What is a reasonable and safe range of temperatures? This range may vary with the size and (decorative) shape of the balloon. All balloons have a temperature gauge in the crown. Suppose that 58 readings (for a balloon in equilibrium) gave a mean temperature of x = 97°C. For this balloon, σ ≈ 17°C. (a) Compute a 95% confidence interval for the average temperature at which this balloon will be in a steady-state equilibrium. (Round your answers to one decimal place.) lower limit      °C upper limit      °C

How hot is the air in the top (crown) of a hot air balloon? Information from Ballooning: The Complete Guide to Riding the Winds, by Wirth and Young (Random House), claims that the air in the crown should be an average of 100°C for a balloon to be in a state of equilibrium. However, the temperature does not need to be exactly 100°C. What is a reasonable and safe range of temperatures? This range may vary with the size and (decorative) shape of the balloon. All balloons have a temperature gauge in the crown. Suppose that 56 readings (for a balloon in equilibrium) gave a mean temperature of x = 97°C. For this balloon, ? ≈ 16°C. (a) Compute a 95% confidence interval for the average temperature at which this balloon will be in a steady-state equilibrium. (Round your answers to one decimal place.) lower limit      °C upper limit      °C (b) If the average temperature in the crown of the balloon goes above the high end of your confidence interval, do you expect that the balloon will…

A4