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NSWMaths Extension 1Quick questions

Statistical Analysis (ME-S1)

Quick questions on The binomial distribution: definition, probability mass function, mean and variance

14short 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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Let XX be the number of successes in nn independent Bernoulli trials, each with success probability pp. Then XX has the binomial distribution B(n,p)B(n, p) (or Bin(n,p)\text{Bin}(n, p)).
What is the probability mass function?
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$P(X=k)=(nk)pk(1−p)n−k.P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}.$
What is mean and variance?
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By summing nn independent Bernoulli trials (each with mean pp and variance p(1−p)p(1 - p)):
What is conditions for the binomial model?
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1. A fixed number of trials, nn. 2. Each trial has exactly two outcomes (success or failure).
What is symmetric and skewed cases?
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The skew decreases as nn grows; for large nn, the binomial approaches the normal (see the normal approximation dot point).
What is pmf at a specific value?
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A multiple-choice quiz has 55 questions, each with 44 options. If you guess randomly, what is the probability of getting exactly 33 correct?
What is conditions check?
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A urn has 55 red and 55 blue marbles. Draw 44 without replacement. Is the number of red draws binomial?
What is verifying the model?
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In a factory, machines produce items independently. Each item has probability 0.020.02 of being defective. In a sample of 100100 items, find the mean and standard deviation of the number defective.
What is two-trial product?
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Two independent flips of a coin with p=0.7p = 0.7. Find P(X=1)P(X = 1).
What is linear transformation?
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Y=3X+5Y = 3 X + 5 where X∼B(10,0.5)X \sim B(10, 0.5). Find E(Y)E(Y) and Var(Y)\text{Var}(Y).
What is confusing nn and kk?
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nn is the fixed number of trials; kk is the (variable) number of successes.
What is forgetting the binomial coefficient?
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(nk)\binom{n}{k} accounts for the ways the kk successes can be arranged.
What is independence assumption broken?
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Sampling without replacement breaks independence and changes the model to hypergeometric.
What is using the formula when pp varies between trials?
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If pp is not constant, the model is not binomial. :::

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