STAT2001 / STAT5008: Compute the Total Quantity of Petroleum Commercially: Mathematical Statistics Assignment, ANU, Australia

Subject STAT2001 / STAT5008: Mathematical Statistics

OBJECTIVE

Compute the total quantity of petroleum commercially recoverable from two fields.

BACKGROUND INFORMATION

Let X be a random variable, define 𝑃(alpha) as 𝑃[x> p(alpha)]=𝛼. For example, if 𝑋~𝑁(200, 10^2), p0.9 is defined as 𝑃[x>p0.9]=0.9. A simple calculation will show that 𝑃0.9= 187.1845

In the petroleum industry, 𝑃0.9 is denoted as P90 and is considered as proved reserves value for practical purposes.

Let xa and 𝑋b be the quantity of petroleum commercially recoverable from field A and field B with the parameters defined as below

Let xa and 𝑋b be the quantity of petroleum commercially recoverable from field A and field B with the parameters defined as below

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PART I

(a) Compute π‘ƒπŸ—πŸŽπ‘¨ value for fields A defined as 𝑃[𝑋𝑨 >π‘ƒπŸ—πŸŽπ‘¨]= 0.9
(b) Compute π‘ƒπŸ—πŸŽπ‘© value for fields B defined as 𝑃[𝑋𝑩 >π‘ƒπŸ—πŸŽπ‘©]= 0.9
(c) Assuming fields A and B are independent, specify the distribution of 𝑋a+ 𝑋b and compute its parameter values. Compute π‘ƒπŸ—πŸŽπ‘¨+B, where 𝑃[𝑋𝑨 + xB >π‘ƒπŸ—πŸŽπ‘¨]= 0.9.
(d) Assuming 𝑋a and 𝑋b have correlation 𝜌, Specify the distribution of 𝑋a + 𝑋b and compute its parameter values. Compute π‘ƒπŸ—πŸŽ a+b for various correlation values. Plot your results with a range of correlation values from (‐1, 1) on the x-axis and corresponding values of π‘ƒπŸ—πŸŽπ‘¨+B on the y-axis. Compare the values of P90a+ P90b with that of π‘ƒπŸ—πŸŽπ‘¨+B.

PART II

(a) Explain how the feedback from part I is incorporated in this submission. Your submission will not be marked if you do not submit an answer to this question.

(b) An oil and gas company, PetroMath is thinking of purchasing both fields A and B. Based on the above results write a short report (max 100 words + figure) for the business manager of PetroMath providing useful information to help with a business decision.

(c) Research the terms β€œValue‐at‐Risk” and β€œrisk diversification”. Explain how the above results apply in this context…

(d) Manager of PetroMath thinks that your assumption of normality may not be valid, however, the mean and variance may remain the same. It is anticipated that the distribution for fields A and B are log‐normal and gamma respectively. How would this assumption change the results? Write an update for the manager (max 50 words+ figure optional) to make sure you are highlighting the key differences in the new results.

 

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