12/1/2019 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences Outline Probabilities summary Distributions Advanced Quantitative Methods in Herd Management Discrete distributions Continuous distributions The Normal Distribution and Confidence Density functions Intervals The normal distribution Leonardo de Knegt • Distribution functions Anders Ringgaard Kristensen • Sampling • Hypothesis testing • Confidence intervals Slide 2 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences Distributions Summary of probabilities • In some cases the probability is defined by a certain function defined over the sample space. • Probabilities may be interpreted • As frequencies • In those cases, we say that the outcome is drawn from a standard distribution. • As objective or subjective beliefs in certain events • The belief interpretation enables us to represent uncertain • There exist standard distributions for many natural phenomena. knowledge in a concise way. • If the sample space is a countable set, we denote the corresponding distribution as discrete. Slide 3 Slide 4 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences Continuous distributions Discrete distributions • If X is the random variable representing the outcome, the expected value of a In some cases, the sample space S of a distribution is not discrete distribution is defined as countable. • If, furthermore, S is an interval on R, the random variable X taking values in S is said to have a continuous distribution. • For any x S , we have P( X = x ) = 0. = 1 ∗ 1 6 + 2 ∗ 1 6 + 3 ∗ 1 6 + 4 ∗ 1 6 + 5 ∗ 1 6 + 6 ∗ 1 𝐹 𝑌 6 = 3.5 • Thus, no probability function exists for a continuous distribution can’t be expressed in tabular form. The variance is defined as • Instead, the distribution is characterized by a density function f ( x ). 1 � + 2 � + 3 � + 4 � + 5 � + 6 � − 3.5 2 = 2.93 𝑊𝑏𝑠 𝑌 = 6 Slide 5 Slide 6 1
12/1/2019 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences Density functions Continuous distributions The density function f has the following properties (for a , b R and a b ) • For a continuous distribution, the expected value E( X ) is defined as And the variance is (just like the discrete case) • Thus, for a continuous distribution, f can only be interpreted as a probability when integrated over an interval. Slide 7 Slide 8 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences The normal distribution The normal distribution • • If S = R, and the random variable X has a normal distribution on S , then The normal distribution may be used to represent almost all kinds of the density function is random outcome on the continuous scale in the real world. • Exceptions: phenomena that are bounded in some sense (e.g. the waiting time to be served in a queue cannot be negative) • It can be showed (central limit theorems) that if X 1 , X 2 , …, X n are random variables of (more or less) any kind, then the sum Y n = X 1 + X 2 + The expected value and the variance simply turn out to be E( X ) = , and • Var( X ) = 2 …+ X n is normally distributed for n sufficiently large. • The normal distribution is the cornerstone among statistical • We say that X is N( , 2 ), or X ~ N( , 2 ) distributions. Slide 9 Slide 10 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences Normal distributions Normal distributions • The normal distribution with = 0, and = 1 is called the Three normal distributions standard normal distribution. 0,5 • A random variable being standard normally distributed is 0,4 often denoted as Z m=0, s=3 0,3 f( x ) m=-5, s=1 0,2 m=0, s=1 • The density function of the standard normal distribution is 0,1 often denoted as (phi) . It follows that 0 -10 -5 0 5 10 x Three normal distributions with mean m and standard deviation s Slide 11 Slide 12 2
12/1/2019 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences Normal distributions: operation properties Normal distributions Let X 1 ~ N( 1 , 1 2 ), X 2 ~ N( 2 , 2 • 2 ), and X 1 and X 2 are independent. From the previous slide it follows in particular, that if X ~ N( , 2 ), then • • Define Y 1 = X 1 + X 2 and Y 2 = X 1 − X 2 . Then • Y 1 ~ N( 1 + 2 , 1 2 + 2 2 ) • Y 2 ~ N( 1 − 2 , 1 2 + 2 2 ) So, if f is the density function of X ~ N( , 2 ), then • • Let a and b be arbitrary real numbers, and let X ~ N( , 2 ). • Define Y = aX + b . Then, Y ~ N( a + b, a 2 2 ) Thus, we can calculate the value of any density function for a normal distribution from the density distribution of the standard normal distribution. Slide 13 Slide 14 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences Distribution functions Distribution functions Even though the definition is the same, the value of the distribution • function is calculated in different ways for the two classes of We have so far defined distributions by: distributions. • probability functions (discrete distributions) • density functions (continuous distributions). • For discrete distributions • We might just as well have used the distribution function F , which is defined in the same way for both classes of distributions: F ( x ) = P( X x ) • • For continuous distributions • F(x) is called the distribution function, or the Cumulative Distribution Function (CDF) Slide 15 Slide 16 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences Distribution functions Distribution functions Recap: • F(X): Cumulative Distribution Function of X • F(x) = P(X ≤ x) • From the formula above, it follows directly that for a continuous distribution, F ’( x ) = f ( x ) • probability that the random variable X takes a value smaller than some specific value x • The distribution function of the standard normal distribution is often denoted as , and naturally ’( z ) = ( z ) . No closed form (formula) exists for , it must be looked up in tables. • f(x): Probability Density Function of x • • Probability that the random variable x takes a value within an interval Slide 17 Slide 18 3
12/1/2019 Department of Veterinary and Animal Sciences Department of Veterinary and Animal Sciences Distribution functions Sampling form a distribution Any distribution function F has the following two properties: Assume that X 1 , X 2 , …, X n are sampled independently from the same distribution having the known expectation and the known standard • F ( x ) 0 for x - deviation • F ( x ) 1 for x Then the mean of the sample Three normal distributions Three normal distributions 0,5 1 0,4 0,8 m=0, s=3 m=0, s=3 0,3 0,6 f( x ) f( x ) m=-5, s=1 m=-5, s=1 has the expected value and the standard deviation 0,2 0,4 m=0, s=1 m=0, s=1 0,1 0,2 0 0 -10 -5 0 5 10 -10 -5 0 5 10 x x Density function Distribution function In particular, if the X i ’s are N( , 2 ) then the sample mean is N( , 2 / n ) Slide 19 Slide 20 Department of Veterinary and Animal Sciences Sampling from a Normal Distribution Hypothesis testing in normal distributions A normal distribution with standard deviation 3 A normal distribution with standard deviation 3 • Assume that X 1 , X 2 , …, X n are sampled independently from the same 1 0,2 0,8 0,6 normal distribution N( , 2 ) where is unknown and is known. f( x ) f( x ) 0,1 m=0, s=3 m=0, s=3 0,4 0,2 0 0 -1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 0 0 x x For some reason we expect (hope) that has a certain value 0 , and we • • Observations close to the mean are far more likely than distant would therefore like to test the following hypothesis: observations. H 0 : = 0 • From the distribution function we can calculate the likelihood that an observation falls within the interval ± • How can we do that? Well, we know that the sample mean is N( , 2 / n ) Slide 21 Hypothesis testing in normal distributions Hypothesis testing in normal distributions 𝑎 = 𝑌 − 𝜈 For a standard normal distribution (z): P (𝜈 − 𝑌 𝜈 + ) = (𝜈 + ) - (𝜈 − ) 𝜏 P (0 − 1 𝑌 0 + 1) = (0 + 1 ) − (0 − 1 ) 𝑔(𝑦) = ( ��� � ) …which that translates to our density function: P (−1 𝑌 1) = (1 ) - (−1 ) Look up at the z table! Phi is the area For the interval ± , where = 0, =1 • under the curve! P (𝜈 − 𝑌 𝜈 + ) = (𝜈 + ) - (𝜈 − ) F(x) = P(X ≤ x) Always gives the area to the left - 0 4
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