Suppose that X_{1}, X_{2}, ..., X_{200} is a set of independent and identically distributed Gamma random variables with parameters alpha = 4, lambda = 3. Describe the normal distribution that would correspond to the sum of those 200 random variables if the Central Limit Theorem holds.

Question
Random variables
Suppose that $$X_{1}, X_{2}, ..., X_{200}$$ is a set of independent and identically distributed Gamma random variables with parameters $$\alpha = 4, \lambda = 3$$.
Describe the normal distribution that would correspond to the sum of those 200 random variables if the Central Limit Theorem holds.

2021-02-27
Step 1
We have been given $$X_{1}, X_{2}, X_{3},…………..X_{200}$$ are independently and identically distributed gamma random variables with $$\prop = 4\ and\ \lambda = 3$$.
Thus, we have the mean and variance for gamma distribution is given as.
$$Mean = \mu = \alpha \lambda = 4\times 3=12$$
$$Variance = sigma^{2}=\alpha \lambda^{2}=4\times 3^{2}=36$$.
Step 2
Since, the random sample is quite large enough of size 200 and also the random variables are independently and identically distributed.
Hence, according to the central limit theorem, the sum of these random variables has normal distribution with mean and standard deviation which is given as below.
Where, $$X = X_{1}+X_{2}+X_{3}+……..X_{200}$$
$$Mean(X)=E(X_{1}+X_{2}+X_{3}+........X_{200})$$
$$=E(X_{1})+E(X_{2})+......E(X_{200})$$
$$=12\times 200=2400$$
Step 3
$$Variance(X)=V(X_{1}+X_{2}+X_{3}+.....X_{200})$$
$$=V(X_{1})+V(X_{2})+....V(X_{200})$$ (since $$X_{1}X_{2}.....X_{200}$$ are independently distributed)
$$=36\times 200=7200$$
Thus, the normal distribution that would correspond to the sum of those 200 random variables if the Central Limit Theorem holds is
$$X\sim N(2400,7200)$$

Relevant Questions

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Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.
1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
1
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