Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Continuous Probability Distributions Chapter 4 The Normal Distribution 1 Introduction A company that makes hot chocolate mix has a machine packages the mixes. The amount in each package varies slightly from the nominal amount. For instance, the 16 oz containers actually contain between 15.95 and 17 oz of hot chocolate. If the discrepancies are distributed uniformly over the interval, what is the probability that a package is underfilled? The Normal Distribution 2 The Uniform Distribution A very simple continuous probability distribution is the uniform distribution. If a random variable X varies uniformly between a and b, then X~U(a,b). The pdf of X is flat, as shown at the right. The Normal Distribution 3 The Uniform Distribution What are f(x), F(x), E(X), and Var(X)? What is the probability that the hot chocolate filler machine underfills the 16oz container? The Normal Distribution 4 Introduction to the Normal Distribution Chapter 5 The Normal Distribution 5 Introduction MENSA is an organization that “provides intelligent individuals an opportunity to meet other smart people at the local, regional, and national levels.” The MENSA website does not specify what IQ score a person must obtain to be admitted, instead it states: “The society welcomes people from every walk of life whose IQ is in the top 2% of the population”. What IQ score must I have to be admitted to MENSA? The Normal Distribution 6 Introduction Average IQ is about 100 points Standard deviation is about 15 points IQ is approximately normally distributed The Normal Distribution 7 The Normal Distribution A normally distributed random variable (also called Gaussian) X~N(μ,σ2) has pdf 1 f ( x) = e 2π σ −( x−μ )2 2σ 2 for -∞≤x≤∞. E(X)=μ Var(X)=σ2 The Normal Distribution 8 The Normal Distribution The normal distribution has very broad applicability in part because • Phenomena in the natural world that result from the interaction of many environmental and genetic factors tend to follow the normal distribution (e.g., height, weight, measurable intelligence). • The sums and averages of random samples have distributions that look roughly normal – as the sample size gets larger, the normal approximation gets better. This result is known as the Central Limit Theorem. The Normal Distribution 9 The Normal Distribution The Standard Normal Distribution: N(0,1) 1 f ( x) = φ ( x) = e 2π − x2 2 x F ( x) = Φ ( x) = ∫ φ ( y )dy −∞ In general, if X~N(μ,σ2), then Z=(X- μ)/σ ~ N(0,1). The Normal Distribution 10 The Normal Distribution Z=(X- μ)/σ ~ N(0,1) is a “standardized” version of the random variable X. The probability values of a general normal distribution can be related to the cumulative distribution function of the standard normal distribution Φ(X) as follows: ⎛b−μ ⎞ ⎛a−μ⎞ P ( a ≤ X ≤ b) = Φ⎜ ⎟ − Φ⎜ ⎟ ⎝ σ ⎠ ⎝ σ ⎠ Table I in the book can be used to calculate probability values of the standard normal distribution. The Normal Distribution 11 The Normal Distribution 12 The Normal Distribution If X~N(2.5 , 9) and Z~N(0, 1), find the following probabilities: – – – – P(Z ≤ 1) P(X ≥ 7) P(1.4 ≤ Z ≤ 2.5) P(|X|≤ 3) What is the probability that a randomly selected individual has an IQ less than 70? The Normal Distribution 13 The Normal Distribution Table I can also be used to find percentiles of the standard normal distribution. Example: If X~N(2.5 , 9) and Z~N(0, 1), find the value of x or z that satisfies each equation. – P(X ≤ x) = .45 – P(Z ≥ z) = .39 – P(|Z| ≥ z ) = .72 What is the minimum score a person must obtain to be admitted to MENSA? (Recall: MENSA admits the top 2% of the population). The Normal Distribution 14 The Normal Distribution The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 266 days and standard deviation 16 days. What is the probability that a pregnancy lasts less than 240 days (about 8 months)? What is c for which there is a 99% probability that a pregnancy lasts longer than c days? The Normal Distribution 15 The Normal Distribution The probability that a normal random variable takes a value within 1 standard deviation of its mean is about 68%, within 2 standard deviations is about 95%, and within 3 standard deviations is about 99.7%. P(1 ≤ Z ≤ 1) ≈ .68 P(2 ≤ Z ≤ 2) ≈ .95 P(3 ≤ Z ≤ 3) ≈ .997 The Normal Distribution 16 The Normal Distribution For α<0.5, the (1- α)x100th percentile of the distribution is denoted by Zα, such that Φ(Zα)=1-α. These percentiles, Zα, are sometimes known as “critical points”. There is a probability of 1-α that a N(μ,σ2) random variable takes on a value within the interval [μ-σ Zα/2, μ+σ Zα/2]. We know from the previous example that if X=“the length of a human pregnancy from conception to birth”, then X~N(266, 256). Since Z0.005 = 2.576, we can be 99% confident that the duration of the pregnancy of a randomly selected woman will be in the interval [μ-σ Z0.005, μ+σ Z0.005]. The Normal Distribution 17 Combining Normal R.V.s We can use combinations of normal random variables to address more complicated questions such as: The mean pregnancy length, 266 days, is about .73 years. What is the probability that a randomly selected woman who gives birth 4 times will spend more than three years of her life (1095 days) pregnant? The Normal Distribution 18 Combining Normal R.V.s If X is normally distributed, then a linear function of X also has a normal distribution. •If a and b are constants and X~N(μ,σ2) and Y=aX+b, then Y~N(aμ+b, a2σ2). •If X1~N(μ1,σ12) and X2~N(μ2,σ22) are independent random variables, then Y=X1+X2~N(μ1+ μ2,σ12+ σ22). [This can be extended to n independent normal random variables.] •If Xi~N(μ,σ2), 1≤ i ≤n, are independent random variables, ⎛ σ2 ⎞ then ⎟⎟ X ~ N ⎜⎜ μ , n ⎠ ⎝ The Normal Distribution 19 Combining Normal R.V.s Example: Suppose the amount of soda pop in a ‘16 oz’ bottle is normally distributed with mean 16 oz and standard deviation 0.25 and that the amounts in the bottles are independent. What is the probability that there will be less than 95 oz of soda in 6 bottles? What are the mean and the standard deviation of the average amount of soda pop in 6 bottles? The Normal Distribution 20 Combining Normal R.V.s An animal shelter has a special facility for sick animals. Suppose there are 3 cats and 2 dogs in the shelter and all at risk for either “cat flu” or “dog flu”. Let X=“the duration of cat flu (hours)” and Y=“the duration of dog flu (hours)”. If X~N(36,9) and Y~N(41, 6.25): ~ What is the distribution of Z=“the total number of pet days spent in the infirmary”? ~ What is the probability that the average time a cat spends in the infirmary is more than 40 hours? The Normal Distribution 21 Approximating Distributions with the Normal Distribution The Normal Distribution 22 Approximating Distributions with the Normal Distribution Recall: if X~B(n,p), then E(X)=np and Var(X)=np(1-p). A binomial distribution, B(n,p), can be approximated by a normal distribution with the same mean and variance, N(np,np(1-p)). The Normal Distribution 23 Approximating Distributions with the Normal Distribution A continuity correction is used to improve the approximation. If X~B(np,np(1-p)), then ⎛ x + 0.5 − np ⎞ ⎟ P ( X ≤ x ) ≈ Φ⎜ ⎜ np(1 − p) ⎟ ⎝ ⎠ and ⎛ x − 0.5 − np ⎞ ⎟ P ( X ≥ x ) ≈ 1 − Φ⎜ ⎜ np(1 − p) ⎟ ⎝ ⎠ These approximations work well for np≥5 and n(1-p) ≥5. The Normal Distribution 24 Approximating Distributions with the Normal Distribution Example: If X~B(14, .4), find P(X ≤ 3) exactly and by using the normal approximation. Example: In a test for a particular illness, a false-positive result is obtained about 1 in 125 administrations of the test. If the test is administered to 15,000 people, estimate the probability of there being more than 135 false-positive results. The Normal Distribution 25 Approximating Distributions with the Normal Distribution The Central Limit Theorem: If X1,…Xn, is a sequence of independent identically distributed random variables with a mean μ and variance σ2, then the distribution of their average is approximately normal ⎛ σ2 ⎞ X ~ N ⎜⎜ μ , ⎟⎟ n ⎠ ⎝ The distribution of the sum X=X1+…+Xn is also approximately normal ( X ~ N nμ , nσ The Normal Distribution 2 ) 26 Approximating Distributions with the Normal Distribution Example: The number of cracks in a ceramic tile has a Poisson distribution with parameter λ=2.4. Use the normal approximation to find the probability that there are more than 1250 cracks in 500 ceramic tiles. The Normal Distribution 27 Distributions Related to Normal The Chi-Square Distribution: If X ~ N(0,1) then Y = X2 has a chi-square distribution with 1 degree of freedom. The Normal Distribution 28 Distributions Related to Normal A t-distribution with ν degrees of freedom is defined as follows: tν = N (0,1) χν /ν 2 where the N(0,1) and Xv2 are independently distributed. As v→∞, the t-distribution tends toward a standard normal distribution. The Normal Distribution 29 Distributions Related to Normal The F-distribution is defined as the ratio of two independent chisquare random variables divided by their respective degrees of freedom: Fν 1ν 2 ~ χν χν 2 ν1 2 ν2 1 2 The Normal Distribution 30