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NSWMaths Extension 1Quick questions

Statistical Analysis (ME-S1)

Quick questions on Bernoulli trials: definition, parameters, mean and variance

11short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is definition?
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A Bernoulli trial is a random experiment with exactly two possible outcomes, conventionally labelled "success" (value 11) and "failure" (value 00), where success occurs with probability pp and failure with probability q=1−pq = 1 - p.
What is mean (expected value)?
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$E(X)=1⋅p+0⋅(1−p)=p.E(X) = 1 \cdot p + 0 \cdot (1 - p) = p.$
What is variance?
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$E(X2)=12⋅p+02⋅(1−p)=p.E(X^2) = 1^2 \cdot p + 0^2 \cdot (1 - p) = p.$
What is why this matters?
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A Bernoulli trial is the simplest non-trivial random variable. The binomial distribution counts the number of successes in a fixed number of independent Bernoulli trials.
What is maximising variance?
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For what value of pp is Var(X)=p(1−p)\text{Var}(X) = p(1 - p) maximised?
What is sum of independent Bernoullis?
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Three independent fair coin flips. Let SS be the total number of heads. Find E(S)E(S) and Var(S)\text{Var}(S).
What is linearity check?
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Two independent Bernoulli trials with p1=0.4p_1 = 0.4 and p2=0.6p_2 = 0.6. Let S=X1+X2S = X_1 + X_2. Find E(S)E(S) and Var(S)\text{Var}(S).
What is mean is pp, not 1p\frac{1}{p}?
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1p\frac{1}{p} is the mean number of trials until the first success (geometric distribution), which is a different model.
What is forgetting independence?
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Sum of two Bernoullis X1+X2X_1 + X_2 has variance equal to the sum of variances only if they are independent.
What is confusing Bernoulli with binomial?
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A single Bernoulli trial has parameter pp alone; a binomial has parameters nn (count) and pp (success probability).
What is range of pp?
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p∈[0,1]p \in [0, 1]. If a calculation gives p<0p < 0 or p>1p > 1, something is wrong. :::

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