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
asked 2021-02-26
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.

Answers (1)

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
\(X\sim N(2400,7200)\)
0

Relevant Questions

asked 2020-12-30
Consider a set of 20 independent and identically distributed uniform random variables in the interval (0,1). What is the expected value and variance of the sum, S, of these random variables?
Select one:
a.\(\displaystyle{E}{\left({S}\right)}={10}\), \(\displaystyle{V}{a}{r}{\left({S}\right)}={1.66667}\)
b.\(\displaystyle{E}{\left({S}\right)}={200}\), \(\displaystyle{V}{a}{r}{\left({S}\right)}={16.6667}\)
c.\(\displaystyle{E}{\left({S}\right)}={100}\), \(\displaystyle{V}{a}{r}{\left({S}\right)}={200}\)
a.\(\displaystyle{E}{\left({S}\right)}={10}\), \(\displaystyle{V}{a}{r}{\left({S}\right)}={12}\)
asked 2020-11-22
Independent random variables \(\displaystyle{X}_{{{1}}}{\quad\text{and}\quad}{X}_{{{2}}}\) are combined according to the formula \(\displaystyle{L}={3}\cdot{X}_{{{1}}}+{2}\cdot{X}_{{{2}}}\).
If \(\displaystyle{X}_{{{1}}}{\quad\text{and}\quad}{X}_{{{2}}}\) both have a variance of 2.0, what is the variance of L?
asked 2021-02-13
if \((X_{1},...,X_{n})\) are independent random variables, each with the density function f, show that the joint density of \((X_{(1)},...,X_{(n)}\) is n!f \((x_{1})..f(x_{n}), x_{1}
asked 2021-01-19
Which possible statements about the chi-squared distribution are true?
a) The statistic X^2, that is used to estimate the variance S^2 of a random sample, has a Chi-squared distribution.
b) The sum of the squares of k independent standard normal random variables has a Chi-squared distribution with k degrees of freedom.
c) The Chi-squared distribution is used in hypothesis testing and estimation.
d) The Chi-squared distribution is a particular case of the Gamma distribution.
e)All of the above.
asked 2020-12-07
Suppose you take independent random samples from populations with means \(\displaystyle\mu{1}{\quad\text{and}\quad}\mu{2}\) and standard deviations \(\displaystyle\sigma{1}{\quad\text{and}\quad}\sigma{2}\). Furthermore, assume either that (i) both populations have normal distributions, or (ii) the sample sizes (n1 and n2) are large. If X1 and X2 are the random sample means, then how does the quantity
\(\displaystyle\frac{{{\left(\overline{{{x}_{{1}}}}-\overline{{{x}_{{2}}}}\right)}-{\left(\mu_{{1}}-\mu_{{2}}\right)}}}{{\sqrt{{\frac{{{\sigma_{{1}}^{{2}}}}}{{{n}_{{1}}}}+\frac{{{\sigma_{{2}}^{{2}}}}}{{{n}_{{2}}}}}}}}\)
Give the name of the distribution and any parameters needed to describe it.
asked 2021-01-16
A population of values has a normal distribution with \(\displaystyle\mu={200}\) and \(\displaystyle\sigma={31.9}\). You intend to draw a random sample of size \(\displaystyle{n}={11}\).
Find the probability that a sample of size \(\displaystyle{n}={11}\) is randomly selected with a mean less than 226.9.
\(\displaystyle{P}{\left({M}{<}{226.9}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-10-28
A population of values has a normal distribution with \(\displaystyle\mu={200}\) and \(\displaystyle\sigma={31.9}\). You intend to draw a random sample of size \(\displaystyle{n}={11}\).
Find the probability that a single randomly selected value is less than 226.9.
\(\displaystyle{P}{\left({X}{<}{226.9}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-12-25
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
6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. 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?
9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) 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?
asked 2021-02-08
A population of values has a normal distribution with \(\displaystyle\mu={204.3}\) and \(\displaystyle\sigma={43}\). You intend to draw a random sample of size \(\displaystyle{n}={111}\).
Find the probability that a single randomly selected value is less than 191.2.
\(\displaystyle{P}{\left({X}{<}{191.2}\right)}=\)?
Find the probability that a sample of size \(\displaystyle{n}={111}\) is randomly selected with a mean less than 191.2.
\(\displaystyle{P}{\left({M}{<}{191.2}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-02-19
A population of values has a normal distribution with \(\displaystyle\mu={216.9}\) and \(\displaystyle\sigma={87.1}\). You intend to draw a random sample of size \(\displaystyle{n}={97}\).
Find the probability that a single randomly selected value is between 193 and 244.3.
\(\displaystyle{P}{\left({193}{<}{X}{<}{244.3}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
...