Statistical Treatment of Experimental Data - Revision history

Admin: /* Chapter 4: Important Probability Distributions */

2017-11-30T04:20:09Z

Chapter 4: Important Probability Distributions

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	This chapter covers the following distributions:		This chapter covers the following distributions:
			* uniform distribution - dice etc.
	* binomial distribution - used for trials (binary outcomes)		* binomial distribution - used for trials (binary outcomes)
	* poisson distribution - used for distribution of outcomes that are positive numbered		* poisson distribution - used for distribution of outcomes that are positive numbered
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	* F distribution - used in tests involving comparison fo two distribution variances (ANOVA)		* F distribution - used in tests involving comparison fo two distribution variances (ANOVA)
	* distribution of sample mean and sample variance for normal case - important extension of discussion of normal distribution		* distribution of sample mean and sample variance for normal case - important extension of discussion of normal distribution

			===Uniform distribution===

			When a "fair" process (such as a six-sided die) occurs, it has a uniform distribution.

			In general, a variable x can be between a and b, in the interval:

			<math>
			[a,b)
			</math>



	===Binomial distribution===		===Binomial distribution===

Admin: /* Exponential distribution */

2017-11-29T16:53:01Z

Exponential distribution

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	<math>		<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	\mu &=& E(X) = int_{0}^{\infty} a x e^{-ax} dx \\		\mu &=& E(X) = \int_{0}^{\infty} a x e^{-ax} dx \\
	&=& \left[ x e^{-ax} + \int e^{-ax} dx \right]_{0}^{\infty} \\		&=& \left[ x e^{-ax} + \int e^{-ax} dx \right]_{0}^{\infty} \\
	&=& \dfrac{1}{a}		&=& \dfrac{1}{a}

Admin: /* Chi squared distribution */

2017-11-10T01:32:52Z

Chi squared distribution

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	Example of this type of random variable: sum of squares of n independent standard normal variates, distributed as <math>\chi_{n}^2</math>		Example of this type of random variable: sum of squares of n independent standard normal variates, distributed as <math>\chi_{n}^2</math>

	Equivalently, if <math>X_1, X_2, \dots, X_n</math> independent random variables, each distributed as <Math>N(\mu, \sigma^2)</math>, then <math>\dfrac{ \sum ~~\left~~(X_i - \mu)^2 }{ \sigma^2 }</math> distributed as <math>\chi_n^2</math>		Equivalently, if <math>X_1, X_2, \dots, X_n</math> independent random variables, each distributed as <Math>N(\mu, \sigma^2)</math>, then <math>\dfrac{ \sum (X_i - \mu)^2 }{ \sigma^2 }</math> distributed as <math>\chi_n^2</math>

	It can also be shown that <math>\dfrac{ \sum (X_i - \overline{X})^2 }{ \sigma^2 } = \dfrac{(n-1)s^2}{\sigma^2}</math> is also distributed as <math>\chi_{n-1}^2</math>, independently of <math>\overline{X}</math>		It can also be shown that <math>\dfrac{ \sum (X_i - \overline{X})^2 }{ \sigma^2 } = \dfrac{(n-1)s^2}{\sigma^2}</math> is also distributed as <math>\chi_{n-1}^2</math>, independently of <math>\overline{X}</math>

Admin: /* Chi squared distribution */

2017-11-10T01:32:34Z

Chi squared distribution

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	Shorthand: <math>X \sim \chi_{\nu}^2</math>		Shorthand: <math>X \sim \chi_{\nu}^2</math>

			Example of this type of random variable: sum of squares of n independent standard normal variates, distributed as <math>\chi_{n}^2</math>

			Equivalently, if <math>X_1, X_2, \dots, X_n</math> independent random variables, each distributed as <Math>N(\mu, \sigma^2)</math>, then <math>\dfrac{ \sum \left(X_i - \mu)^2 }{ \sigma^2 }</math> distributed as <math>\chi_n^2</math>

			It can also be shown that <math>\dfrac{ \sum (X_i - \overline{X})^2 }{ \sigma^2 } = \dfrac{(n-1)s^2}{\sigma^2}</math> is also distributed as <math>\chi_{n-1}^2</math>, independently of <math>\overline{X}</math>

			If <math>X \sim \chi_{\nu}^2</math>, the mean and variance are given by:

			Mean: <math>E(X) = \nu</math>

			Variance: <math>V(X) = 2 \nu</math>

			Additive property: if X1 and X2 are independently distributed as <math>\chi_{\nu_1}^2</math> and <math>\chi_{\nu_2}^2</math>, then their sum X1+X2 is distributed as <math>\chi_{\nu_1 + \nu_2}^2</math>

	==Flags==		==Flags==

Admin: /* Chi squared distribution */

2017-11-10T01:28:07Z

Chi squared distribution

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	</math>		</math>

	Shorthand: <math>X \sim \~~Chi_~~{\nu}^2</math>		Shorthand: <math>X \sim \chi_{\nu}^2</math>

	==Flags==		==Flags==

Admin: /* Chi squared distribution */

2017-11-10T01:27:54Z

Chi squared distribution

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	f(x) = \dfrac{ x^{\frac{1}{2}\nu - 1} e^{-\frac{1}{2} x} }{ \Gamma( \frac{1}{2} \nu) 2^{\frac{1}{2} \nu} } \qquad \nu > 0, 0 \leq x < \infty		f(x) = \dfrac{ x^{\frac{1}{2}\nu - 1} e^{-\frac{1}{2} x} }{ \Gamma( \frac{1}{2} \nu) 2^{\frac{1}{2} \nu} } \qquad \nu > 0, 0 \leq x < \infty
	</math>		</math>

			Shorthand: <math>X \sim \Chi_{\nu}^2</math>

	==Flags==		==Flags==

Admin: /* Normal distribution example */

2017-11-10T01:27:30Z

Normal distribution example

← Older revision		Revision as of 01:27, 10 November 2017
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	&=& 0.1974		&=& 0.1974
	\end{array}		\end{array}
			</math>


			===Chi squared distribution===

			Random variable X is distributed as chi squared with <math>\nu</math> degrees of freedom if density given by:

			<math>
			f(x) = \dfrac{ x^{\frac{1}{2}\nu - 1} e^{-\frac{1}{2} x} }{ \Gamma( \frac{1}{2} \nu) 2^{\frac{1}{2} \nu} } \qquad \nu > 0, 0 \leq x < \infty
	</math>		</math>

Admin: /* Normal distribution example */

2017-11-10T01:22:53Z

Normal distribution example

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	P(X > 3.5) &=& P(Z > 0.25) \\		P(X > 3.5) &=& P(Z > 0.25) \\
	&=& 1 - \Phi(0.25) \\		&=& 1 - \Phi(0.25) \\
	&=& 1 - 0.5987 = 0.413		&=& 1 - 0.5987 \\
			&=& 0.413
	\end{array}		\end{array}
	</math>		</math>
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	\begin{array}{rcl}		\begin{array}{rcl}
	P(X < 1.2) &=& P(Z < -0.9) \\		P(X < 1.2) &=& P(Z < -0.9) \\
	&=& \Phi(-0.9) = 1 - \Phi(0.9) \\		&=& \Phi(-0.9) \\
			&=& 1 - \Phi(0.9) \\
	&=& 1 - 0.8159 \\		&=& 1 - 0.8159 \\
	&=& 0.1841		&=& 0.1841
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	P(2.5 < X < 3.5) &=& P(-0.25 < Z < 0.25) \\		P(2.5 < X < 3.5) &=& P(-0.25 < Z < 0.25) \\
	&=& \Phi(0.25) - \Phi(-0.25) \\		&=& \Phi(0.25) - \Phi(-0.25) \\
	&=& 2 \Phi(0.25) - 1 = 1.1974 - 1 \\		&=& 2 \Phi(0.25) - 1 \\
			&=& 1.1974 - 1 \\
	&=& 0.1974		&=& 0.1974
	\end{array}		\end{array}

Admin: /* Normal distribution example */

2017-11-10T01:22:20Z

Normal distribution example

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	<math>		<math>
	\begin{array}{rc}		\begin{array}{rcl}
	P(X > 3.5) &=& P(Z > 0.25) \\		P(X > 3.5) &=& P(Z > 0.25) \\
	&=& 1 - \Phi(0.25) \\		&=& 1 - \Phi(0.25) \\
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	<math>		<math>
	\begin{array}{rc}		\begin{array}{rcl}
	P(X < 1.2) &=& P(Z < -0.9) \\		P(X < 1.2) &=& P(Z < -0.9) \\
	&=& \Phi(-0.9) = 1 - \Phi(0.9) \\		&=& \Phi(-0.9) = 1 - \Phi(0.9) \\

Admin: /* Normal distribution example */

2017-11-10T01:22:00Z

Normal distribution example

← Older revision		Revision as of 01:22, 10 November 2017
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	Q3: What is probability of observing 2.5 < X < 3.5?		Q3: What is probability of observing 2.5 < X < 3.5?

	Question 1: convert X to Z by plugging in to definition: <math>Z = \dfrac{X - \mu}{\sigma} = \dfrac{X - 3}{2}</math>.		Question 1: convert X to Z by plugging in to definition: <math>Z = \dfrac{X - \mu}{\sigma} = \dfrac{X - 3}{2}</math>. Now:

	~~Now,~~ <math>P(X > 3.5) = P(Z > 0.25) = 1 - \Phi(0.25) = 1 - 0.5987 = 0.413</math>		<math>
			\begin{array}{rc}
			P(X > 3.5) &=& P(Z > 0.25) \\
			&=& 1 - \Phi(0.25) \\
			&=& 1 - 0.5987 = 0.413
			\end{array}
			</math>

	Question 2: again, convert X to Z.		Question 2: again, convert X to Z. Now:

	~~Now~~ <math>P(X < 1.2) = P(Z < -0.9) = \Phi(-0.9) = 1 - \Phi(0.9) = 1 - 0.8159 = 0.1841</math>		<math>
			\begin{array}{rc}
			P(X < 1.2) &=& P(Z < -0.9) \\
			&=& \Phi(-0.9) = 1 - \Phi(0.9) \\
			&=& 1 - 0.8159 \\
			&=& 0.1841
			\end{array}
			</math>

	Question 3: convert from X to Z, which gives		Question 3: convert from X to Z, which gives

	<math>		<math>
	P(2.5 < X < 3.5) = P(-0.25 < Z < 0.25) = \Phi(0.25) - \Phi(-0.25) = 2 \Phi(0.25) - 1 = 1.1974 - 1 = 0.1974		\begin{array}{rcl}
			P(2.5 < X < 3.5) &=& P(-0.25 < Z < 0.25) \\
			&=& \Phi(0.25) - \Phi(-0.25) \\
			&=& 2 \Phi(0.25) - 1 = 1.1974 - 1 \\
			&=& 0.1974
			\end{array}
	</math>		</math>