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Statistics

Quick questions on Statistical distributions: the binomial and Normal models and standardising - OCR A-Level Maths A

7short 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 are discrete random variables?
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A discrete random variable takes separate values, each with a probability; the probabilities must sum to 11. The mean (expected value) is E(X)=∑x P(X=x)E(X) = \sum x\,P(X = x), the long-run average value.
What is the binomial distribution?
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The binomial X∼B(n,p)X \sim B(n, p) counts the successes in nn independent trials, each with the same success probability pp.
What is the inverse Normal?
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When a probability is given and a value is wanted, work backwards: find the zz-value for that probability (the inverse Normal), then convert to XX with X=μ+zσX = \mu + z\sigma. Two such conditions give simultaneous equations for an unknown μ\mu and σ\sigma.
What is choosing a binomial model?
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Before using the binomial, check the conditions hold. A fixed number of independent trials with a constant success probability fits; sampling without replacement from a small population does not, because pp changes between trials.
What is the Normal approximation to the binomial?
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When nn is large and pp is not too close to 00 or 11, the binomial B(n,p)B(n, p) is approximately Normal with the same mean and variance, N(np,np(1−p))N(np, np(1 - p)). Because you replace a discrete distribution with a continuous one, apply a continuity correction (for example P(X≥50)P(X \ge 50) becomes P(X>49.5)P(X > 49.5)).
What is q1?
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X∼B(8,0.25)X \sim B(8, 0.25). Find the mean and variance. [2 marks]
What is q2?
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X∼N(50,16)X \sim N(50, 16). Find P(X<54)P(X < 54). [2 marks]

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