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# STA 3032 Module 1-2 Outline STA3032

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This 10 page Study Guide was uploaded by Tia Belvin on Saturday January 23, 2016. The Study Guide belongs to STA3032 at University of Florida taught by Demetris Athienitis in Spring 2016. Since its upload, it has received 58 views. For similar materials see Engineering Statistics in Engineering and Tech at University of Florida.

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Date Created: 01/23/16

STA3032 MODULE 1 ARTICLE I. 1.1-1.3.5 1) 1.1 Descriptive Statistics a) Population parameters- “a numerical summary concerning the complete collection of subjects” i) Notated by Greek characters b) Sample statistics- “a numerical summary concerning a subset of the population, from which we try to draw inference about the population parameter” i) Notated by the hat symbol over a population parameter c) i.e. σσstandard deviation 2) 1.2 Summary Statistics 3) 1.2.1 Location a) Mode: most frequent number b) Mean: average of the observations c) p percentile: p% of data are less than that specific value and 100%-p% are greater i) position of p percentile value: (p/100)(n+1) (1) i.e. value x was in the 90 percentile which means that 90% of all valuethwere less than value x d) Median: 50 percentile e) Trimmed Mean: a% trimmed mean means that a% multiplied by your total numerical values= some value that is equal to how many values you “trim” from the lower and upper side of your data (assuming your data is ordered smallest to largest). With the remaining values you recalculate the new average. i) i.e. if you have 10 values, ordered, and you want to calculate the 10% trimmed mean. 10%=0.1 so 0.1 x 10 (your total number of values) =1. Therefore, you would trim off one value from the lower side and one value from the upper side. With these 8 remaining values you would recalculate the mean and that is your “trimmed mean”. NOTE: The mean is more sensitive to outliers than the median. 4) 1.2.2 Spread a) Variance: a measure of spread of observations from their center (mean). n 2 1 2 2 i) σ σ ([ ] )−nx n−1 i=1 b) Standard Deviation: the square root of the varianceσ=σ c) Range: maximum observation- minimum observation d) Interquartile Range (IQR): 75 percentile-25 percentile (Q3-Q1) 5) 1.3 Graphical Summaries 6) 1.3.1 Dot Plot a) Stack each observation on a horizontal line. This gives an idea of the shape of the data. 7) 1.3.2 Histogram a) Step 1: Create class intervals (you choose boundary points) b) Step 2: Construct a frequency table c) Step 3: Draw a rectangle for each class 8) 1.3.3 Box-Plot a) A box plot only uses quartiles. b) You need to find Q ,1Q ,2Q a3, the IQR given your data. c) Constructing the graph i) Left vertical line: Your lowest value within 1.5IQR of Q ( i1 it is not within 1.5IQR of Q 1t is considered an outlier) ii) Make a box connecting your: (1) Lower whisker: your Q va1ue (2) Upper whisker: your Q va3ue iii) The median is your Q a2d should be the vertical line within the box iv) Right vertical line: Your highest value within 1.5IQR of Q (i3 it is not within 1.5IQR of Q 3t is considered an outlier) NOTE: outliers are marked with a dot and your min value should have a horizontal line connecting it to the first quartile as should your maximum value have a horizontal line connecting it to the third quartile 9) 1.3.4 Pie Chart a) Size of slice is determined by fraction of the 360˚ that corresponds to that category. b) Each slice is labeled and color coordinated to your choosing. 10) 1.3.5 Scatter Plot a) Used to plot points of two variables in order to display a relationship between the two MODULE 2 ARTICLE II. 2.1-2.7 2.1 SAMPLE SPACE AND EVENTS 1) Sample Space (S): set of all possible outcomes a. i.e. for rolling a die (S) = {1,2,3,4,5,6} 2) Event: subset of a sample space a. i.e. let X be the event of an odd outcome when rolling a die, then X={1,3,5} 2.1.2 RELATING EVENTS 1) When there are multiple events within the sample space we can use Venn Diagrams a. Combining events: A B = all listings in sample space that are either A or B or both b. Intersecting events: A B = all listings in the sample space that are only both A and B c. Complement of an event: A = all listings in sample space that are not A d. Mutually exclusive: A and B are mutually exclusive if A B = 0 2.2 PROBABILITY 1) P(A) denotes the probability that A occurs a. P(S)= 1 b. 0 ≤ P(A)≤ 1 c. If events A and B are mutually exclusive then P(A B)= P(A)+P(B) d. P(A )= 1-P(A) e. P(A)=k/N i. N being the equally likely outcomes of S ii. k being A’s outcomes (k≤N) f. P(A B)= P(A) + P(B) – P(A B) 2.3 COUNTING METHODS 1) Product rule: “if the first task of an experiment can result 1n n possible outcomes and for such outcome, the second task can result in n p2ssible outcomes, and so forth up to k tasks, then the total number of ways to k perform the sequence of k tasks is ∏ i=1(n i” 2.3.1 PERMUTATIONS 1) Permutation: the number of ordered arrangements of r objects selected from n objects (without replacement) and r ≤ n 2.3.2 COMBINATIONS 1) Combination: number of unordered arrangements (without replacement) and r ≤ n 2.4 CONDITIONAL PROBABILITY AND INDEPENDENCE 1) Unconditional probability: a probability based upon entire sample space 2) Conditional probability: a probability based upon a subset of a sample space a. The conditional probability of A given B 3) Rule of Multiplication a. If P(A) does not equal 0 then P(A B)= P(B|A)P(A) b. If P(B) does not equal 0 then P(A B)= P(A|B)P(B) 2.4.1 INDEPENDENT EVENTS 1) 2 events A and B are independent if the probability of each remains the same whether or not the other has occurred a. If either P(A)=0 or P(B)=0, the 2 events are independent b. If both are not equal to zero then i. P(B|A)= P(B) ii. P(A|B)= P(A) 2.4.2 LAW OF TOTAL PROBABILITY 1) If A1,….Anare mutually exclusive and exhaustive events, and B is any event, then A i P(¿∩B) n P ( ) ∑ ¿ i=1 2) If P(Ai does not equal zero for each Aithen B∨A i P(¿)P(Ai n P ( ) ∑ ¿ i=1 2.4.3 BAYES RULE 1) Usually P(B|A) does not equal P(A|B), Bayes rule is a way to calculate one knowing the other c a. P(A), P(A ), and P(B) cannot equal zero 2.5 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1) Random variable: function that assigns a numerical value to each outcome of an experiment a. Discrete: a countable set of possible values i. Probability mass function (p.m.f.) 1. P(X=x)=px(x) (for random variable X) 2. 0 ≤ p(x) ≤ 1 3. The summation of all probabilities is equal to 1 b. Continuous: unaccountably infinite set of possible values i. Probability density function (p.d.f.) 1. Denoted by x (x) 2. P(X=x)=0 3. f(x) ≥ 0 4. the integral from -∞ to ∞ of f(x)dx is equal to 1 2.5.1 EXPECTED VALUE AND VARIANCE 1) The expected value or mean of random variable X is ∞ E(X =μ x ∫f (x)dx=∑ xp(x) −∞ 2) The variance of a random variable X is V X =σ=E2 (X )−E(X) x 2.5.2 POPULATION PERCENTILES 1) Let X be a continuous random variable a. The population p percentile, x is p xp F(xp)= ∫ f() dt=p/100 −∞ 2.5.3 CHEBYSHEV’S INEQUALITY 1) Let X be a random variable a. E(X)=µ b. V(X)=σ 2 1 P (X−μ |<kσ )1− 2 k 2) This implies that the probability that a random variable differs from its mean by standard deviations or more is never greater than 1/k 2.5.4 JOINTLY DISTRIBUTED RANDOM VARIABLES 1) Jointly Distributed: “When two or more random variables are associated with each item in a population” b d P(a<X<b∧c<Y<d )=∫∫ f x,y dydx a c 2.5.5 CONDITIONAL DISTRIBUTIONS 1) Let X and Y be jointly continuous random variables with joint p.d.f. f(x,y). Let fx(x) denote the marginal p.d.f. of X and let x be any number for whixh f (x) is greater than zero 2) The conditional p.d.f. of Y|X = x is f (x,y) fY∨Xy|x)= fX(x) 2.5.6 INDEPENDENT RANDOM VARIABLES 1) Let X and Y be two continuous random variables, they are independent if f(x,y)=f Xx)f Yy) 2.5.7 COVARIANCE 1) Population covariance is “a measure of strength of a linear relationship among two variables” Cov(X,Y= E(XY)−E X)E(Y) 2) If X and Y are independent then E(XY= E(X)E(Y) ¿thecovariaweoulequal ¿ ) 3) Population correlation is also “a measure of the strength of the linear relationship “ a. Varies from covariance because it is unitless and ranges from -1 to 1 2.5.9COMMON DISCRETE DISTRIBUTIONS 1) Bernoulli only has two possible outcomes a. Success (X=1) with probability p b. Failure (X=0) with probability 1-p c. Properties of bernoulli: x 1−x px)=p(1−p) E(X)=μxp 2 V(X)=σxp(1−p) 2) Binomial corresponds to n Bernoulli trials a. The trials are independent b. Each trial has an identical probability p c. The random variable X is the total number of successes d. Properties of Binomial: E(X)=μxnp 2 V X =σ=xp (1−p) 3) Geoa. Deals with the number of Bernoulli trials necessary to achieve the first b. Properties of Geometric: p(x=p(1−p) x−1 4) Negative Binomial a. Extension of geometric th b. It is the number of trials up to and including the s success c. Properties of NB: E X =μ x s p 2 V X =σ=x1−p)/p 2 5) Poisson a. Occurs when we count the number of occurrences of an event over a given interval of time or space b. Properties of Poisson: λ= np p(x=λe /x! E(X)=μ= λ x 2 V(X)=σ=x c. The estimate of λ is λ = X/t λσ d. With estimated uncertaintyσσλ= √ t 2.5.10 COMMON CONTINUOUS DISTRIBUTIONS 1) A uniform random variable places equal weight to all values within its support and is continuous. a. A uniform random variable has p.d.f. 1 fx)= for a≤x≤b b−a 2) Normal a. The normal distribution has parameter µ and a scale parameter σ b. Normal distribution has p.d.f. −1(x−μ) fx)= 1 e 2σ2 for -∞<x<∞ σ√2π 2.6 CENTRAL LIMIT THEOREM 1) CLT has three versions: classical, Lyapunov, and Linderberg a. All say that the asymptotic distribution of the sample mean X is normal 2 X N(μ, σ ) for n>30 n i. Binomial: np>5 and n(1−p >5 ii. Poisson: λ>10 2.7 NORMAL PROBABILITY PLOT 1) The probability plot is a graphical technique for comparing two data sets, either two empirical, or one empirical and one theoretical. 2) The empirical c.d.f. is the cumulative distribution function associated with the empirical measure of the sample. F (x) n 3) There are two types of plots used to plot the empirical c.d.f. to the normal theoretical one ( G(x)¿ F a. A P-P plot plots (¿¿n x),G(x)) ¿ FF b. A Q-Q plot plots the quantile functions −1 −1 (¿¿n (x),G (x) ) ¿

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