Probability and Statistics Flashcards by Evie Mayhew

What is the binomial coefficient

(n k) = n! / k!(n-k)!

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what is a Bernoulli trial

only has 2 possible outcomes

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rule for probability that 2 independent events both occur

‘and’ rule -> multiplication

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rule for probability that one or another event occurs

addition

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how to find probability that a and b occur given that b occurs

P(a and b) / P(b)

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pdf for binomial distribution
P(X=k) =

(n k) p^k (1-p)^n-k

binomial coefficient X probability of success k times X probability of failure n-k times

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definition of Expectation

the sum of all possible outcomes, weighted by their probabilities

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When can the Poisson distribution be used

large n
small p
(ie rare events)

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formula for µ, the density parameter

µ = np (=E(x))
n = number of trials
p = probability of success

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pdf for poisson distribution P(X=k) ≈

e^µ µ^k / k!

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E(x) for binomial distribution

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E(x) for Poisson distribution

µ = np

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what are the parameters for the geometric distribution

p, probability of success

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pdf for geometric distribution
P(X = k) =

(1 - p)^n-1 p

probability of the n-1 failures before the one probability of success

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expectation E(x) for geometric distribution

1 / p

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parameters for the exponential distribution

lambda = the rate parameter

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what’s the difference between the exponential and geometric distribution

geometric = discrete
exponential = continuous
exponential distribution can be used to model the geometric when n gets large and p gets very small

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pdf for exponential distribution
f(x) =

lambda e^ - (lambda x)

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cdf for exponential distribution
F(x) =

1 - e^ - (lambda x)

(if can’t remember can integrate the pdf between 0 and x)

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what does the cdf show

an expression that gives the probability that a random variable X falls between 0 and x

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expected value of the exponential distribution

1 / lambda

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parameters of the normal distribution

µ - the mean
sigma - std

expectation for normal distribution

µ = mean

what does the Z scale do (normal distribution)

measures how many stds a point lies from the mean of its parent distribution

normalises the data

formula for Z scale

Z = (Xi - µ) / std X = point µ = mean of parent distribution std = std of parent distribution

critical value for 2 tailed standard normal at alpha=0.05

+- 1.96 +-1.96*sigma for not normalised

when is the t distribution used

small sample size don't know mean

difference between t distribution and normal distribution

t has longer tale, therefore has more extreme critical values for same significance level as the sample size in t increases, the t distribution tends to the normal

formula for t scale

( X - µ) / Sx X = sample mean µ = population mean (often unknown) Sx = standard error of mean

what is standard error of mean (SEM)

Sx = s / root(n) s = sample std n = sample size

what is the p value

probability of observing a result equal to or more extreme than the outcome

what is a type one error

rejecting the null when its true 'False positive'

what is a type two error

fail to reject the null when its false 'False negative'

what is alpha level

level of confidence at which we reject the null probability of a type one error

why shouldn't you use multiple t tests for multiple comparisons

the probability of a type 1 error gets large

what should you use instead of multiple t tests for comparisons

ANOVA

what is the within-group variance

comparing the distribution of replicates to their treatment mean

what is the among/between group variance

comparing the distribution of the treatment means to the grand mean

what is the F statistic in ANOVA

among / within

what are treatments in ANOVA

the different samples

what are replicates in ANOVA

sample units within treatments

formula for Chi-square test statistic

∑ (o - e)^2 / e

formula for Pearsons r test statistic

(use Z scale) r = ∑(Zxi + Zyi) / n-1

formula for slope estimate, b of a regression line

b = ∑(Xi - X)(Yi - Y) ---------------------------- ∑(Xi - X)(Xi - X) Xi = x values X = mean of x values

residual formula

residual = Yi - ^Yi y value minus the value of y on the regression line

problems with regression analysis

- induced correlations ( ie values that sum to 100% or 1, such as mineral compositions may indicate correlation in more than one variable falsely) - correlation vs causation - pseudoreplication (single area data taken from doesn't represent all)

What is the t-test used for

- test whether a sample is drawn from a population of specific mean - test if means of 2 samples differ

what is the ANOVA test used for

- test whether ≥ 3 samples are drawn from populations with equal means (like students t)

what is the Chi-square test used for

- test how well observed categorial data fits a given model/expected values

How to find within-group variance

s.s / d.f s.s. = ∑(Xi - X)^2 -> distance from treatment means d.f. = n-1 (for each treatment, then added together (ie total replicates - number of treatments))

how to find among (between) group variance

s.s / d.f s.s. = ∑ (Xti - Xg) -> distance of treatment means from grand mean d.f. = n - 1 ( number of treatments -1)

when do you reject ANOVA null hypothesis

when F statistic > table value, based on numerator and denominator degrees of freedom

assumptions for t test

- data from normally distributed populations - data from populations of equal variance - samples drawn at random from parent distributions

assumptions for ANOVA

- data drawn from normally distributed populations - data from populations of equal variance - data independent of one another