Question

1. Chain Rule and the Markov Rule

1. You as a sports analyst are modeling the free throw success of Lebron James. You are asked to predict the probability that Lebron will make four free throws in serial succession during an important game that takes place in the opposite team’s venue.

Based on your data of Lebron’s previous free throws, you estimate that the median probability (0.5 below and 0.5 above) of a single free throw by Lebron under a wide range of expected conditions is 0.75. Also, estimate that the probability of free throw 2 given he made free throw 1 is P(2|1) = 0.8. Also you estimate that the Pr of free throw 3 given he made free throw 2 with kudos is P(3|2) = 0.9. But because of the extreme conditions at the important game in the opposite team’s site, you estimate that the Pr he makes free throw 4 given he made free throw 3 is 0.7.

Using these data, write the expression for and predict to 2 sd the Pr of success of four free throws in succession. Assume that the Pr he makes the first free throw is his median value of 0.75. As in b) assume the Markov Rule in which the primary dependency is with the previous free throw and further back free throws are less influential and considered independent for this analysis. Also, calculate the mean value and variance, and standard deviation of the probabilities where each probability value is equally weighted. Finally write an expression and calculate the numerical value of P(1,2,3,4) to two standard deviations, which represents uncertainty management of estimating the epistemic uncertainty of the probability values.

               Mean Probabilities=

Variance =

                Standard Deviation =  

P(1,2,3,4) =      to two standard deviations, which represents uncertainty management of epistemic uncertainty in the estimated probability values.

1. b. Now predict the probability to 2 sd of Lebron’s making 4 free throws in series if each free throw is considered independent of all of the other free throw attempts. Use Lebron’s median Pr value of success = 0.75.

P(1,2,3,4) =

Assume the same standard deviation calculated in a:

P(1,2,3,4) =        to two standard deviations exhibiting an estimate of epistemic uncertainty in the analysis. Do the results for c) and d) agree within the combined estimated uncertainties in the two values?

.

Answer 1

Here, we are to predict the probability that Lebron will make four free throws in serial succession.

Based on the data on Lebron’s previous free throws, we estimate that

P(Lebron successfully makes a free throw)=0.75

P(2|1)=P(Lebron successfully makes 2 free throws given he had successfully made 1 free throw)=0.8,

P(3|2)=P(Lebron successfully makes 3 free throws given he had successfully made 2 free throws)=0.9, and

P(4|3)=P(Lebron successfully makes 4 free throws given he had successfully made 3 free throws)=0.7

Based on these, we need to find P(1,2,3,4)=P(Lebron successfully makes 4 free throws successively)

Part a:

Let X_i be the random variable denoting that Lebron successfully makes the i^th free throw, i=1,2,3,4,... i.e,

Here, we note that, for Lebron to successfully make 2 throws, he has to successfully make both the first and the second throws, i.e., X₁=1 and X₂=1

Again, for Lebron to successfully make 3 throws, he has to successfully make all of the first, the second and the third throws, i.e., X₁=1, X₂=1 and X₃=1, and so on.

Then, we get

Thus, we have,

Part b:

Here, we assume that the sequence {X_n} of random variables is a Markov Chain on the state space S={0,1}

Thus, we have,

Now, having obtained the two estimates, we get

Mean Probabilty=0.3682

Variance of the two probabilities=0.00009604

Standard Deviation of the two probabilities=0.0098

The Standard deviation being very low, the average probability serves as a good estimate to the required probability

Part c:

Here, we assume that the X_i's are independently distributed, for every i=1,2,3,....

Hence, our required probability is given by