Probstat/notes/chi-squared distribution - ประวัติรุ่นแก้ไข

Jittat: /* Definition */

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Definition

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Sample variances

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Sample variances

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Sample variances

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Sample variances

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Sample variances

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Sample variances

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Sample variances

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Sample variances

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 == Definition ==
+{{กล่องฟ้า|
-Let <math>Z_1,Z_2,\ldots,Z_n</math> be independent unit normal random variables.  A random variable
+'''Definition:''' Let <math>Z_1,Z_2,\ldots,Z_n</math> be independent unit normal random variables.  A random variable
 <center>
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 <math>X\sim \chi_n^2.</math>
 </center>
 Wikipedia has a [http://en.wikipedia.org/wiki/Chi-squared_distribution nice article on chi-squared distribution] which also includes plots of its pdf and cdf.

@@ แถว 9: / แถว 9: @@
 </center>
-is called a ''chi-squared'' random variable with <math>n</math> degree of freedom.  We also write
+is called a ''chi-squared'' random variable with <math>n</math> degrees of freedom.  We also write
 <center>
@@ แถว 23: / แถว 23: @@
 === Expectations and variances ===
-If <math>X</math> is chi-squared with <math>n</math> degree of freedom, we have that its expectation
+If <math>X</math> is chi-squared with <math>n</math> degrees of freedom, we have that its expectation
 <center>
@@ แถว 50: / แถว 50: @@
 </center>
-Note that the only differences between this term and the previous one is the use of sample mean <math>\bar{X}</math> instead of the population mean.  It can be proved (not here) that in fact, for normal population, <math>S^2</math> (with some scaling) is actually a chi-squared random variable with <math>n-1</math> degree of freedom (instead of <math>n</math> as in the term computed from <math>\mu</math>).  That is,
+Note that the only differences between this term and the previous one is the use of sample mean <math>\bar{X}</math> instead of the population mean.  It can be proved (not here) that in fact, for normal population, <math>S^2</math> (with some scaling) is actually a chi-squared random variable with <math>n-1</math> degrees of freedom (instead of <math>n</math> as in the term computed from <math>\mu</math>).  That is,
 <center>

@@ แถว 50: / แถว 50: @@
 </center>
-Note that the only differences between this term and the previous one is the use of sample mean <math>\bar{X}</math> instead of the population mean.  It can be proved (not here) that in fact, for normal population, <math>S^2</math> (with some scaling) is actually a chi-squared random variable with <math>n-1</math> degree of freedom (instead of <math>n</math> as in the previous term).  That is,
+Note that the only differences between this term and the previous one is the use of sample mean <math>\bar{X}</math> instead of the population mean.  It can be proved (not here) that in fact, for normal population, <math>S^2</math> (with some scaling) is actually a chi-squared random variable with <math>n-1</math> degree of freedom (instead of <math>n</math> as in the term computed from <math>\mu</math>).  That is,
 <center>

@@ แถว 50: / แถว 50: @@
 </center>
-Note that the only differences between this term and the previous one is the use of sample mean <math>\bar{X}</math> instead of the population mean.  It can be proved (not here) that in fact, for normal population, <math>S^2</math> (with some scaling) is actually a chi-squared random variable with <math>n-1</math> degree of freedom.  That is,
+Note that the only differences between this term and the previous one is the use of sample mean <math>\bar{X}</math> instead of the population mean.  It can be proved (not here) that in fact, for normal population, <math>S^2</math> (with some scaling) is actually a chi-squared random variable with <math>n-1</math> degree of freedom (instead of <math>n</math> as in the previous term).  That is,
 <center>

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 More over, for normal population, <math>\bar{X}</math> and <math>S^2</math> are independent.  This is very surprising because both quantities are calculated from the same sample.
 == Links ==
 * [http://en.wikipedia.org/wiki/Chi-squared_distribution Wikipedia article on the chi-squared distribution]

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 The chi-squared distribution is very important when we want to reason about the sample variances (See [[Probstat/notes/sample means and sample variances|notes]]).   Under that settings, we sample <math>X_1,X_2,\ldots,X_n</math> from a normal population whose mean is <math>\mu</math> and variance is <math>\sigma^2</math>.
-Recall that <math>(X_i - \mu)/\sigma</math> is unit normal.  Therefore,
+We first note that if <math>\mu</math> and <math>\sigma^2</math> is known, a term similar to the one we used to compute the same variance is chi-squared.  Recall that <math>(X_i - \mu)/\sigma</math> is unit normal.  Therefore,
 <center>
 <math>\sum_{i=1}^n \left(\frac{X_i - \mu}{\sigma}\right)^2 = \frac{1}{\sigma^2}\sum_{i=1}^n (X_i - \mu)^2 \sim \chi_n^2.</math>
 </center>
 == Links ==
 * [http://en.wikipedia.org/wiki/Chi-squared_distribution Wikipedia article on the chi-squared distribution]