Further Mathematics · OCR H245 · Option S · Topic 5.02

Discrete random variables

Probability distributions for general discrete random variables, the binomial distribution, the discrete uniform distribution, the geometric distribution, the Poisson distribution.

  • 5 sub-topics
  • 4 in Year 12, 1 in Year 13
  • 119 practice questions
  • 5 animated lessons
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What the specification says

Every sub-topic OCR can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. 5.02a-cProbability distributions for general discrete random variablesY12
  2. 5.02dThe binomial distributionY12
  3. 5.02eThe discrete uniform distributionY12
  4. 5.02f-hThe geometric distributionY13
  5. 5.02i-nThe Poisson distributionY12

Try 2 questions

Easy and medium questions from this topic, each with a worked solutions and the mark for every step. All 119 questions in this topic, free, no account needed.

Poisson and binomial distributions

Easy

The Poisson distribution · Year 12 · 1 mark

  1. X∼Po(3.0)X \sim \mathrm{Po}(3.0). What is P(X=0)\mathrm{P}(X = 0)?
    (1)

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  1. P(X=0)\mathrm{P}(X = 0)=e−λ{}= \mathrm{e}^{-\lambda}.CC: 0.04980.0498

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Discrete probability distributions

Medium

A discrete random variable · Year 12 · 7 marks

The discrete random variable XX has the probability distribution: P(X=1)\mathrm{P}(X = 1)=p, P(X=2){}= p, \ \mathrm{P}(X = 2)=q, P(X=3){}= q, \ \mathrm{P}(X = 3)=0.4, P(X=4){}= 0.4, \ \mathrm{P}(X = 4)=0.1{}= 0.1. Given that E(X)=2.4\mathrm{E}(X) = 2.4,
  1. (a)
    find the values of pp and qq.
    (3)
  2. (b)
    Find Var⁡(X)\operatorname{Var}(X).
    (3)
  3. (c)
    Find Var⁡(1+3X)\operatorname{Var}(1 + 3X).
    (1)
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Part (a)3 marks

  1. The probabilities add to 1.p+q=0.5p + q = 0.5
  2. Use E(X)\mathrm{E}(X)=∑xP(X=x){}= \sum x \mathrm{P}(X = x).p+2q+1.6=2.4p + 2q + 1.6 = 2.4
  3. p=0.2, q=0.3p = 0.2, \ q = 0.3

Part (b)3 marks

  1. ∑x2P(X=x)\sum x^{2} \mathrm{P}(X = x).E(X2)=6.6\mathrm{E}(X^2) = 6.6
  2. Var⁡(X)\operatorname{Var}(X)=6.6−2.42{}= 6.6 - 2.4^2=0.84{}= 0.84

Part (c)1 mark

  1. Var⁡(aX+b)\operatorname{Var}(aX + b)=a2Var⁡(X){}= a^{2} \operatorname{Var}(X).9×0.84=7.569 \times 0.84 = 7.56

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