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GAMMA DISTRIBUTION
GAMMA DISTRIBUTION
GAMMA DISTRIBUTION
GAMMA DISTRIBUTION
The Chi-Square Distribution
The Chi-Square Distribution
The Chi-Square Distribution
The Chi-Square Distribution The relation between Chi-square and Gamma Random variables Chi-square n degrees of freedom ==gamma with parameters n/2 and 1/2
t-Distribution
t-Distribution
This density function is look like normal density function. It is symmetric about ‘0’ and as n becomes larger, it becomes more and more like standard normal density.
t-Distribution Figure shows a graph of the t -density function with 5 degrees of freedom compared with the standard normal density. Notice that the t -density has thicker “tails,” indicating greater variability, than does the normal density.
The mean and variance of Tn can be shown to equal
t-Distribution
t-Distribution
t-Distribution
F-Distribution
F-Distribution
F-Distribution The quantities Fα, n, m are tabulated for different values of n , m, and α ≤ 1/2. If Fα, n, m is desired when α > 1/2 , it can be obtained by using the following relation
THE SAMPLE MEAN The value associated with any member of the population can be regarded as being the value of a random variable having expectation μ and variance σ2. The quantities μ and σ2 are called the population mean and the population variance, respectively. Let X1, X2, . . . , Xn be a sample of values from this population. The sample mean is defined by
THE SAMPLE MEAN
The expected value of the sample mean is the population mean μ whereas its variance is 1/n times the population variance.
The Central Limit Theorem It asserts that the sum of a large number of independent random variables has a distribution that is approximately normal distribution. In its simple form, the theorem is as follows:
The Central Limit Theorem
The Central Limit Theorem Problem: An insurance company has 25,000 automobile policy holders. If the yearly claim of a policy holder is a random variable with mean 320 and standard deviation 540, approximate the probability that the total yearly claim exceeds 8.3 million.
The Central Limit Theorem
Thus, there are only 2.3 chances out of 10,000 that the total yearly claim will exceed 8.3 million.
GAMMA DISTRIBUTION
GAMMA DISTRIBUTION
GAMMA DISTRIBUTION
The Chi-Square Distribution
The Chi-Square Distribution
The Chi-Square Distribution
The Chi-Square Distribution The relation between Chi-square and Gamma Random variables Chi-square n degrees of freedom ==gamma with parameters n/2 and 1/2
t-Distribution
t-Distribution
This density function is look like normal density function. It is symmetric about ‘0’ and as n becomes larger, it becomes more and more like standard normal density.
t-Distribution Figure shows a graph of the t -density function with 5 degrees of freedom compared with the standard normal density. Notice that the t -density has thicker “tails,” indicating greater variability, than does the normal density.
The mean and variance of Tn can be shown to equal
t-Distribution
t-Distribution
t-Distribution
F-Distribution
F-Distribution
F-Distribution The quantities Fα, n, m are tabulated for different values of n , m, and α ≤ 1/2. If Fα, n, m is desired when α > 1/2 , it can be obtained by using the following relation
THE SAMPLE MEAN The value associated with any member of the population can be regarded as being the value of a random variable having expectation μ and variance σ2. The quantities μ and σ2 are called the population mean and the population variance, respectively. Let X1, X2, . . . , Xn be a sample of values from this population. The sample mean is defined by
THE SAMPLE MEAN
The expected value of the sample mean is the population mean μ whereas its variance is 1/n times the population variance.
The Central Limit Theorem It asserts that the sum of a large number of independent random variables has a distribution that is approximately normal distribution. In its simple form, the theorem is as follows:
The Central Limit Theorem
The Central Limit Theorem Problem: An insurance company has 25,000 automobile policy holders. If the yearly claim of a policy holder is a random variable with mean 320 and standard deviation 540, approximate the probability that the total yearly claim exceeds 8.3 million.
The Central Limit Theorem
Thus, there are only 2.3 chances out of 10,000 that the total yearly claim will exceed 8.3 million.
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