正态分布是数理统计中的一个基本概念，同时也是现实中非常常见的一种概率分布，广泛应用于多种场合。比如下面这篇EssayPhD团队原创的统计report代写范文：

Scenario: USASuperCars sells luxury sports cars. It has just signed a contract to sell in New Year's Day (01-01-2016) time, a batch of these super cars to various customers around the globe. The following table shows the orders from eight customers. The selling prices are fixed and in local currencies at the exchange rate prevailing at the time of the delivery. Of course there is uncertainty in the exchange rates, and in order to cope with this uncertainty estimates of the expected value and standard deviation of these have been provided by the Bank of America for all but one (EUR) of the currencies. The report that came with these estimates stated that these rates are normally distributed and independent.

Questions:

以下是问题的解答部分：

Executive Summary

This report concerns the evaluation and risk analysis of the total revenue for USA Super Cars, with the assumption of independent normal distributions of currencies. Two estimates are performed based on the estimation of mean and standard deviation of the currencies from the source and the estimation using the prevailing exchange rates. Using the first method, the mean of the total revenue is $2,191,461 with standard deviation $37,869, and the 5% Value at Risk is $2,129,171. The mean estimated from the second method is $2,529,084 with standard deviation $84,039, and the 5% Value at Risk is $2,390,851. Since the evaluation using the prevailing exchange rates is more reliable, the uncertainty of the risk is larger than expected. Since the bank offers only $2,150,000 for the total revenue and the prevailing exchange rates suggest the company could earn more than this number, the offer is not acceptable if the currency risks are properly hedged. Four ways to hedge the currency risk include investing in currency-hedged mutual fund or exchange-traded fund, shorting the over-valued currency, buying the under-valued currency, and pursuing for higher interest rate. Of course, the decision of the company should incorporate the risk preference and expectation of future changes of exchange rates. In this case, the Sale Manger is risk-averse while the CEO is not. On the other side, the bank if buys the total revenue, then the bank is taking the uncertainties of currencies and the default risk. Actions could be taken after the interested parties taking comprehensive consideration of all sorts of risks. Limitation of the report mainly comes from the assumption of independence and normality, as well as the validity of the estimate of the mean and standard deviation of the currencies.

Introduction

USA Super Cars sells luxury cars and has signed a forward contract to deliver in 1st Jan 2016 to its global customers. Providing that the selling prices are fixed in local currencies at the prevailing exchange rate, a risk analysis is performed to instruct all the instructed parties to cope with the uncertainty. Risk evaluation is performed with the assumed normal distribution of the exchange rate, and the currencies are independent. The report includes the estimation of the revenue and risk analysis for the interested parties for both the estimated exchange rate in 1st Jan 2016 and the prevailing exchange rate. At last limitation and suggestions would be provided based on the data analysis.

Estimation of the revenue

Since the estimation of exchange rate of EUR/USD is not available, it is estimated by using the daily closing values from the last 24 months, 28 November 2013 to 27 November 2015 from Board of Governors of the Federal Reserve System (2015). The mean and standard deviation for Euro is calculated as 1.23213 and 0.11825 to US dollar, respectively. Filling in the estimation of France order, the table of total estimation is listed as in table 1. （EssayPhD 统计report代写范文，请勿转载或用作其他用途）

Since all the currencies are assumed to follow independent normal distribution, the total revenue is actually the linear combination of a series of independent normal random variables. According to Weiss and Weiss (2012), the total revenue follows a normal distribution, with the mean being the sum of means and standard deviation being the square of sum of variances. Specifically, the estimated mean of the total revenue is $2,191,461 and standard deviation is $37,869. That is to say the average tendency of the total revenue is $2,191,461, and the variation to it is $37,869. More intuitively, the distribution of total revenue is plotted in figure 1, as the frequency massed around the mean value. Given the estimated distribution of the total revenue, the probability of the total revenue exceeds or falls below certain threshold could be calculated. Following the cumulative probability density of normal distribution, the revenue that exceeds $2,280,000 has the probability of 0.97% and the revenue that is less than $2,160,000 has the probability of 20.31%.

Risk analysis of interested parties

If HSBC offers to pay $2,150,000 in return of the uncertain currencies, evaluation could be conducted according to the estimated distribution. The cumulative distribution of the total revenue at $2,150,000 is 13.68%, which means the probability of total revenue exceeding $2,150,000 is 13.68% and the probability of total revenue less than this amount is 86.32%. That is to say, there is large chance that the total revenue is worthy of more money than the tendered offer from HSBC. It is not a good offer for USA Super Cars, because the company is very likely to earn more. Furthermore, the offer from HSBC is even less than the mean value of the total revenue, $2,191,461, therefore it could be hardly considered as a good offer.

If the Sales Manager is willing to accept the offer, which means he possesses pessimistic expectation about the future value of the total revenue and the loss hurts him more than the utility from equivalent gain. This fits the definition of risk-averse (Investopedia, 2015) exactly, which refers to the behaviour that the investor prefers to the return with smaller risk and gives up the higher possible return. Since the CEO refuses to accept the offer, that is to say the CEO is expecting a higher return with higher risk and the Sales Manager is more risk-averse. Apart from the uncertainties of the exchange rate, if the offer is accepted by the company the bank is also taking the counterparty risk (Merk Funds, 2014). The forward contract features the risk when the future price moves in favour of the counterparty then the company would have incentive to default, while when the future prices moves against the counterparty then the buyers would default. The latter situation is the counterparty risk, and if the bank’s offer is accepted while the counterparty defaults, this increases the uncertainty of the bank.

Analysis with prevailing exchange rate

If the bank is to pay the money next week, it would better to perform the risk analysis with the prevailing exchange rates. Exporting all the data from Board of Governors of the Federal Reserve System (2015), the standard deviation of the exchange rates are replaced with the estimation using the last 24 month daily data and the mean is replaced with the last closing price in 27 November 2015, as in table 2. The estimated mean of the total revenue is $2,529,084 and standard deviation $84,039, compared to the results $2,191,461 and $37,869 respectively. So if the bank offers to pay $2,150,000, it is not favourable at all since the mean increases to a large number valued by the prevailing exchange rate. Statistically speaking, the probability that the total return would be less than $2,150,000 is 0.00032%, which is extremely unlikely to happen. On the other side, if the bank pays next week, USA Super Cars would be able to secure the revenue in a short time and to reduce the risk. So the situation differs since the risk of the company reduces and the valuation of the total returns changes. When USA Super Cars expects that US dollars depreciate hugely and expects the total return is less than $2,150,000, or the company is extremely risk-averse, the company would prefer the payment to be made next week. Because the payment to be made next week would reduce the uncertainty of the company while the prevailing evaluation of the total revenue suggests the offered price from HSBC is not favourable. When HSBC holds the anticipation that the total revenue in New Year’s Day exceeds the payment plus time value, the bank would prefer the payment to be made next week.（EssayPhD 统计report代写范文，请勿转载或用作其他用途）

Assuming the bank converts all the currencies into US dollars at the prevailing exchange rates, the probability that the bank would incur a loss is 0.00032% as discussed above. As illustrated in figure 2, the distribution of total revenue could be compared in which one is estimated by the provided values while another one uses the prevailing exchange rate. The 5% Value at Risk is $2,390,851 with the estimated normal distribution. Ignoring the time value of the money in this case, the expected profit for the bank is the difference between the mean of revenue and the offer price. Therefore the expected profit is $379,084. The comparison could be made to the situation using the provided values of mean and standard deviation of the currencies, instead of the prevailing exchange rates. The 5% Value at Risk for the valuation of total revenue not using prevailing exchange rates is $2,129,171 and expected profit for the bank in this case is $41,461. Since the provided estimate of mean and deviation of the currencies differs greatly from the estimate using prevailing exchange rates, this is the reason of shift of the Value at Risk and expected profit after using prevailing exchange rates.

As suggested by Borzykowski (2014), there are four ways to hedge to currency risks, including investing in currency-hedged mutual fund or exchange-traded fund, shorting the over-valued currency, buying the under-valued currency, and pursuing for higher interest rate. If the bank chooses not to convert all the currencies or just converts some currencies, those four ways to hedge the currency risks could be adopted.

Limitation

The risk report relies greatly on the assumption of independent normal distributions of currencies and the validity of the estimations. First of all, the independent normal distributions of currencies are not plausible as pointed out in the seminal work of McFarland, Pettit and Sung (1982). The distribution of financial data, especially the foreign exchange rates, features leptokurtosis, which refers to heavy tails and sharp peaks different from normal distribution (Hodrick, 2014). The independent assumption also limits the analysis as pointed out by Bradshaw and Huang (1991) that global currencies are dependent subjecting to structural economic dependences. At last, the estimate of mean and standard deviation of the currencies either provided from the source or using the daily data of last 24 months. The validity of the estimate from the source is hard to evaluate without knowledge of the involved methods, while the estimation using prevailing exchange rates relies on the normal distribution.

Conclusion

As the discussed in the analysis above, the total revenues has the mean value of $2,191,461 and standard deviation $37,869 using the estimate of mean and standard deviation for the currencies provided in the source. If the total revenue is evaluated by the prevailing exchange rates, by replacing the mean with the last daily closing price and the standard deviation with the estimation from daily data of last 24 months, the mean value is $2,529,084 and standard deviation $84,039. Since the standard deviation reflects the risk of the currencies, the company faces higher uncertainty of the currency revenue. Since the bank offers to pay $2,150,000 for the total revenue, the company should be considerate enough to the future trend of foreign currencies and their risk preference. Providing the total revenue is valued using the prevailing exchange rates, the company should not accept the offer from HSBC. Actually, the company could take measures to hedge the risk like invest in currency-hedged mutual fund in order to secure higher revenue.

Reference

Weiss, N. A., & Weiss, C. A. (2012). Introductory statistics. Pearson Education.

Board of Governors of the Federal Reserve System. (2015). Foreign Exchange Rate –H.10. Retrieved from http://www.federalreserve.gov/releases/h10/hist/ on 2 December , 2015.

Investopedia. (2015). Definition of Risk-Averse. Retrieved from http://www.investopedia.com/terms/r/riskaverse.asp on 2 December, 2015.

Merk Funds. (2014). What is the counterparty risk on a forward currency contract? Retrieved from http://www.merkfunds.com/currency-asset-class/currencies-unplugged/what-is-counter-party-risk.html on 2 December 2015.

Borzykowski, B. (2014). 4 ways to protect yourself from foreign-currency risk. Retrieved from http://www.cnbc.com/2014/04/02/4-ways-to-protect-yourself-from-foreign-currency-risk.html on 2 December 2015.

McFarland, J. W., Pettit, R. R., & Sung, S. K. (1982). The distribution of foreign exchange price changes: trading day effects and risk measurement.the Journal of Finance, 37(3), 693-715.

Hodrick, R. (2014). The empirical evidence on the efficiency of forward and futures foreign exchange markets (Vol. 1). Routledge. Bradshaw, Y. W., & Huang, J. (1991). Intensifying global dependency: Foreign debt, structural adjustment, and third world underdevelopment. The Sociological Quarterly, 32(3), 321-342.

这篇EssayPhD团队代写的统计report，由团队创始人roger亲自操刀。Roger来自UCL 统计专业，目前正攻读PhD学位。这说明了我们并非中介，全部是货真价实的PhD团队，因此才能收获客户的信任，才敢接下这个要求保英国80分的统计report代写。