The Graph, Expectation, and Variance of a Binomial Distribution (VCE SSCE Mathematical Methods): Flashcards

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The Graph, Expectation, and Variance of a Binomial Distribution
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Graph shape when p<0.5p < 0.5

Positively skewed (skewed to the right)

Graph shape when p=0.5p = 0.5

Symmetrical

Graph shape when p>0.5p > 0.5

Negatively skewed (skewed to the left)

Expectation formula for binomial distribution

E(X)=npE(X) = np

Variance formula for binomial distribution

extVar(X)=np(1−p) ext{Var}(X) = np(1-p)

What (1−p)(1-p) represents in binomial distribution

Probability of failure

Standard deviation formula for binomial distribution

sigma=sqrtnp(1−p)sigma = sqrt{np(1-p)}

% of outcomes in mupm2sigmamu pm 2sigma interval

Approximately 95%

Relationship between graphs for p=0.2p=0.2 and p=0.8p=0.8

Mirror images of each other

What E(X)=npE(X) = np represents

Long-run average, not guarantee for single trial

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Proofs for the Expectation and Variance

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