SOME SPECIFIC PROBABILITY DISTRIBUTIONS. 1 2πσ. 2 e 1 2 ( x µ

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1 SOME SPECIFIC PROBABILITY DISTRIBUTIONS. Normal random variables.. Probability Density Function. The random variable is said to be normally distributed with mean µ and variance abbreviated by x N[µ, ] if the density function of x is given by f x ; µ, π e x µ The normal probability density function is bell-shaped and symmetric. The figure below shows the probability distribution function for the normal distribution with a µ and. The areas between the two lines is This represents the probability that an observation lies within one standard deviation of the mean. Figure. Normal Probability Density Function.3 Μ, Σ.. Date: August 9, 4.

2 SOME SPECIFIC PROBABILITY DISTRIBUTIONS The next figure below shows the portion of the distribution between -4 and when the mean is one and is equal to two. Figure. Normal Probability Density Function Showing P 4 <x<. Probability Between Limits is Density Critical Value.. Properties of the normal random variable. a: Ex µ, Varx. b: The density is continuous and symmetric about µ. c: The population mean, median, and mode coincide. d: The range is unbounded. e: There are points of inflection at µ ±. f: It is completely specified by the two parameters µ and. g: The sum of two independently distributed normal random variables is normally distributed. If Y α + β + γ where Nµ, and Nµ, and and are independent, then Y Nαµ + βµ + γ; α + β..3. Distribution function of a normal random variable. F x ; µ, Pr x x f s ; µ, ds Here is the probability density function and the cumulative distribution of the normal distribution with µ and.

3 SOME SPECIFIC PROBABILITY DISTRIBUTIONS 3 f Figure 3. Normal pdf and cdf Probability Density Function Cumulative Distribution Function.8 F Evaluating probability statements with a normal random variable. If x Nµ, then, Z µ N, E Z E µ E µ VarZ Var µ Var 3

4 4 SOME SPECIFIC PROBABILITY DISTRIBUTIONS Consequently, Pra x b Pra [ µ x µ b ] µ Pr a µ x µ b µ F b µ ;, F a µ ;, area below 4 Figure 4. Probability of Intervals.3 Μ, Σ.. b Μ Σ a Μ Σ b.96 a.6 We can then merely look in tables for the distribution function of a N, variable..5. Moment generating function of a normal random variable. The moment generating function for the central moments is as follows The first central moment is M t e t. 5 E µ d dt e t t e t 6 The second central moment is

5 SOME SPECIFIC PROBABILITY DISTRIBUTIONS 5 E µ d dt e t d dt t e t t 4 e t + e t 7 The third central moment is E µ 3 d3 dt e t 3 d dt t e 4 t t 3 6 e t t 3 6 e t + + t 4 + 3t 4 e t e t e t + t 4 e t 8 The fourth central moment is E µ 4 d4 dt e t 4 d dt t 3 e 6 t t 4 8 e t t 4 8 e t t 4 + 3t 6 + 6t 6 e t e t e t + 3t 6 e t e t e t 9. Chi-square random variable.. Probability Density Function. The random variable is said to be a chi-square random variable with ν degrees of freedom [abbreviated χ ν ] if the density function of is given by f x ; ν ν Γ v x ν e x < x otherwise where Γ is the gamma function defined by Note that for positive integer values of r, Γr r -! Γr u r e u du r >

6 6 SOME SPECIFIC PROBABILITY DISTRIBUTIONS The following diagram shows the pdf and cdf for the chi-square distribution with parameters ν. Probability Density Function Figure 5. Chi-square pdf and cdf Cumulative Distribution Function..8.8 f.6.4 F Properties of the chi-square random variable.... χ and N,. Consider n independent random variables. It can also be shown that If i N, i,,..., n then n i i χ n If i N, i,,..., n then n i i χ 3 n because this is the sum of n- independent random variables given that and n- of the x s are independent.... χ and Nµ,. If i N µ, i,,..., n n i µ then χ n 4 i n i and χ n i..3. Sums of chi-square random variables. If y and y are independently distributed as χ ν and χ ν, respectively, then y + y χ ν+ν. 5

7 SOME SPECIFIC PROBABILITY DISTRIBUTIONS Moments of chi-square random variables. Mean χ ν ν degrees of freedom Var χ ν ν Mode χ ν ν 6.3. The distribution function of χ ν. F x; ν x f s; νds 7 is tabulated in most statistics and econometrics texts..4. Moment generating function. The moment generating function is as follows The first moment is M t E d dt υ t υ/,t < t υ/ υ t υ + / The Student s t random variable This distribution was published by William Gosset in 98. His employer, Guinness Breweries, required him to publish under a pseudonym, so he chose Student. 3.. Relationship of Student s t-distribution to Normal Distribution. The ratio t N, χ ν ν has the Student s t density function with ν degrees of freedom where the standard normal variate in the numerator is distributed independently of the χ variate in the denominator. Tabulations of the associated distribution function are included in most statistics and econometrics books. Note that it is symmetric about origin. 3.. Probability Density Function. The density of Student s t distribution is given by: f t ; ν Γ ν + πν Γ ν + t ν ν + < t <

8 8 SOME SPECIFIC PROBABILITY DISTRIBUTIONS The following diagram shows the pdf and cdf for the Student s t-distribution with parameter ν. f Figure 6. Student s t distribution pdf and cdf Probability Density Function Cumulative Distribution Function.8 F The following diagram shows the cdf for the Student s t-distribution with parameters ν and ν 3. Figure 7. Student s t-distribution with alternative parameter levels f x v 3.3 v.. 4 4

9 SOME SPECIFIC PROBABILITY DISTRIBUTIONS Moments of Student s t-distribution. Mean t ν Var t ν ν ν 4. The F Fisher variance ratio statistic 4.. Distribution Function. If χ ν and χ ν are independently distributed chi-square variates, then χ ν ν F ν,ν ν χ ν χ ν ν χ ν ν has the F density with ν and ν degrees of freedom Probability Density Function. The density of the F distribution is f F ; ν,ν Γν +ν Γ ν Γ ν otherwise ν ν ν ν F ν +ν + ν ν F F > 4 Tabulations of the distribution of Fν,ν are widely available. Note that F ν, ν and therefore the critical values can be found from f αν,ν f α ν.,ν The following diagram shows the pdf and cdf for the F distribution with parameters ν and ν. F ν,ν f Figure 8. F Distributtion pdf and cdf Probability Density Function Cumulative Distribution Function.8 F Here is the pdf of the F distribution for some alternative values of pairs of values ν and ν.

10 SOME SPECIFIC PROBABILITY DISTRIBUTIONS Figure 9. Probability of Intervals.8 Ν, Ν 5.6 f x.4. Ν, Ν Ν 6, Ν moments of the F distribution. EF ν ν VarF ν ν+ν ν ν ν 4 5 6