The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable

Transkript

1 Probability Distributions for Continuous Random Variables: The Uniform Distribution (Sabit Olas l kl Da¼g l m) The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable f(x) x min x max x Total area under the uniform probability density function is 1.0 1

2 It has propability density function of 8 < f(x) = : 1 x max x min for x min < x < x max 0 elsewhere 9 = ; Its mean is = E(X) = Z xmax x min xf(x)dx = x max + x min 2 2

3 It has propability density function of 8 < f(x) = : Its variance is 1 x max x min for x min < x < x max 0 elsewhere 2 = E[(X ) 2 ] = Z xmax 9 = ; x min (x ) 2 f(x)dx = 1 12 (x max x min ) 2 3

4 Ex: 2 x6 aras nda tan mlanm ş sabit olas l kl da¼g l m düşünelim. Bunun olas l k fonksiyonu 2 x 6 için f(x) = = 0:25 ki şu şekilde gösterilebilir f(x) x 4

5 Ortalamas : = x max + x min 2 = = 4 Varyasyonu: 2 = 1 12 (x max x min ) 2 = 1 12 (6 2)2 = 1:333 5

6 The Normal Distribution (Normal Da¼g l m) The normal distribution is the most important distribution in the statistical theory It is bell-shaped It is symmetrical around the mean Its mean, median and mode are equal Location is determined by the mean, Spread is determined by the standard deviation, 6

7 The random variable has an in nite theoretical range: 1 to +1 f(x) µ s x Mean = Median = Mode By varying the parameters µ and s, we obtain different normal distributions 7

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9 A (normal) random variable is the one having normal distribution where the probability density function is f(x) = 1 p 2 e 1 2 (x )2 for 1 < x < 1; > 0 The normal distribution closely approximates the probability distributions of a wide range of random variables 9

10 Computations of probabilities are direct and elegant Distributions of sample means approach a normal distribution given a large sample size* If random variable X has a normal distribution with and variance 2, then it is shown as X N(; 2 ) 10

11 Cumulative Normal Distribution: When X N(; 2 ), cumulative distribution function is f(x) F (x 0 ) = P (X x 0 ) = Z x0 1 f(x)dx 0 x 0 x 11

12 The probability for a range of values is measured by the area under the curve P (a < X < b) = F (b) F (a) a µ b x 12

13 The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(x) P( < X < μ) = 0.5 P(μ < X < ) = µ P( < X < ) = 1.0 X 13

14 The Standardized Normal (Standart Normal Da¼g l m) Any normal distribution (with any mean and variance combination) can be transformed into the standardized normal distribution (Z), with mean 0 and variance 1 14

15 This need to transform X units into Z units by subtracting the mean of X and dividing by its standard deviation Z = X It obtains the following f(z) Z ~ N(01), 0 1 Z 15

16 Ex: E¼ger X ortalamas 100, standart sapmas 50 olan rassal bir de¼gişken ise, X = 200 de¼gerinin Z karş l ¼g şudur Z = X = = 2 50 Buna göre X = 200 de¼geri X de¼gişkeninin ortalamas olan 100 den 2 standart sapma yüksektedir. Böylece X=200 de¼gerinin, X in alabilece¼gi tüm de¼gerlere için göreli yerini bulmuş oluruz 16

17 Note that the distribution is the same, only the scale is standardized a b x f(x) = < < = < < σ μ a F σ μ b F σ μ b Z σ μ a P b) X P(a σ b μ σ a μ Z µ 0 17

18 The Standardized Normal Table gives probability for any value of z Normal da¼g l ma sahip bir X rassal de¼gişkeni için P(a < X < b) de¼gerini bulal m Önce X in a ve b ye eşit oldu¼gu de¼gerleri Z ye çevirebilir, sonra da kümülatif normal tablosunu kullanabiliriz Ex: X ortalamas 8.0, standart sapmas 5 olan normal da¼g l ma sahip bir rassal de¼gişken olsun (yani X N(8; 25)). P(X < 8.6) de¼gerini 18

19 bulal m Z = X = 8:6 8 5 = 0:12, P (Z < 0:12) = 0:547 µ = 8 s = 10 µ = 0 s = X Z P(X < 8.6) P(Z < 0.12) Yani X rassal de¼gişkeninin alabilece¼gi de¼gerlerin %54.78 i 8.6 n n alt ndad r 19

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21 For upper tail (üst kuyruk) properties Ex: P (Z > 2:00) =? P (Z < 2:00) = 0:9772 ) P (Z > 2:00) = 1 0:9772 = 0:

22 For negative Z-values, use the fact that it is symmetric distribution Ex: P (Z < 2:00) =? P (Z < 2:00) = 0:9772 ) P (Z < 2:00) = 1 0:9772 = 0: Z Z 22

23 Ex: Finding the X value for a Known Probability X N(8; 25) ise X in hangi de¼geri X in alabilece¼gi tüm de¼gerlerin %20 sinin üstündedir?.20.80?

24 Z de¼geri için bahsi geçen de¼gerin 0.84 oldu¼gunu standart normal tablosundan biliyoruz. O halde Z = X ) X = + Z = 8 + ( 0:84)5 = 3:8 24

25 Ex: Araba yedek parças üreten bir şirketin üretti¼gi bir ürünün dayan m süresi normal da¼g l ma sahiptir ve ortalamas 1,250 hafta, standart sapmas da 250 haftad r. Bu ürünlerden rastgele seçilen bir tanesinin 900 ila 1,300 hafta aras nda dayanma olas l ¼g nedir? 25

26 P (900 < X < 1300) = P ( 900 < Z < 1300 ) = P ( < Z < ) = P ( 1:2 < Z < 0:2) = F (0:2) F ( 1:2) = 0:5793 (1 0:8643) = 0:44 26

27 Assessing Normality: Not all continuous random variables are normally distributed. It is important to evaluate how well the data is approximated by a normal distribution. However, there are tests that can be applied, for instance, by the use of statistical programs 27