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# Lecture13 - Lecture 13 Statistics Commonly Used Probability...

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Lecture 13: Statistics • Probability distributions • Example probability calculations Commonly Used Probability Distributions • Gaussian Distribution • Z-score • t-test • Hypergeometric Distribution Gaussian (Normal) Distribution f ( x | μ , " 2 ) = 1 " 2 # e \$ ( x \$ μ ) 2 /(2 " 2 ) μ = mean of population ! 2 = variance of population § Value of Gaussian Distribution : It approximates experimental noise due to independent, random, and additive factors. Can provide a good model of noise in many biological experiments Codon Adaptivity Index GGC AUG GAC CAU AGA GGA CAG UGA C CAI calculation (for E.coli): w = --- 1.0 0.237 0.001 0.004 1.0 CAI = (1 x 0.237 x 0.001 x 0.004 x 1) 1/5 CAI = 0.062 § Classes of codon usage in genes: • Highly expressed genes (CAI > 0.4) • Moderately expressed genes (0.4 > CAI > 0.15) • Low expressed or foreign genes (CAI < 0.15) ---

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Gaussian Example: CAI Query § For a population of yeast genes, what is the probability of obtaining a CAI value greater than 0.8? Strategy: § Calculate the mean ( μ ) and stdev ( ! ) of the yeast gene CAI values § Use the Gaussian distribution to calculate the probability of obtaining a CAI value greater than 0.8: § Can also calculate using Excel function: NORMDIST p = 1 " 1 # 2 \$ e " ( x " μ ) 2 /( 2 # 2 ) dx "% CAI & Distribution of yeast CAI values CAI 0 50 100 150 200 250 Number of Genes 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 • Does this look like a Gaussian Distribution? CAI values μ = 0.199 ! = 0.149 P(CAI > 0.8) = 1 - NormDist( 0.8, 0.199, 0.149, TRUE) P(CAI > 0.8) = 2.7 x 10 -5 Fitting data to a Gaussian Distribution § The skew and and kurtosis of the data indicate how well the data fits a Gaussian distribution.
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