Further Mathematics · OCR H245 · Option S · Topic 5.03

Continuous random variables

Continuous random variables, probability density functions, cumulative distribution functions.

  • 3 sub-topics
  • Taught in Year 13
  • 29 practice questions
  • 3 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. 5.03aContinuous random variablesY13
  2. 5.03b-dProbability density functionsY13
  3. 5.03e-gCumulative distribution functionsY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 29 questions in this topic, free, no account needed.

Continuous probability distributions

Not needed for Year 12Easy

A valid probability density function · Year 13 · 1 mark

  1. Which conditions must a probability density function ff satisfy?
    (1)

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  1. A density can be greater than 1. It is the area, not the height, that is a probability.AA: f(x)≥0f(x) \ge 0 for all xx, and the total area under its graph is 1

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Continuous probability distributions

Not needed for Year 12Medium

Continuous random variables · Year 13 · 5 marks

The continuous random variable XX has probability density function f(x)=kx(2−x)f(x) = kx(2 - x) for 0≤x≤20 \leq x \leq 2, and 0 otherwise.
  1. (a)
    Show that k=34k = \frac{3}{4}.
    (2)
  2. (b)
    Find P(X<23)\mathrm{P}(X < \frac{2}{3}).
    (2)
  3. (c)
    Write down E(X)\mathrm{E}(X), giving a reason.
    (1)

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Part (a)2 marks

  1. The total probability is 1.∫02kx(2−x) dx\int_0^{2} kx(2 - x)\,\mathrm{d}x=1{}= 1
  2. k×43=1k \times \frac{4}{3} = 1

Part (b)2 marks

  1. ∫0233x(2−x)4 dx\int_0^{\frac{2}{3}} \frac{3 x \left(2 - x\right)}{4}\,\mathrm{d}x
  2. =727= \frac{7}{27}

Part (c)1 mark

  1. The pdf is symmetric about its midpoint.11

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Continuous probability distributions

Not needed for Year 12Hard

The cumulative distribution function · Year 13 · 6 marks

The continuous random variable XX has cumulative distribution function F(x)F(x)=0 (x<0), F(x){}= 0 \ (x < 0), \ F(x)=x24 (0≤x≤2), F(x){}= \dfrac{x^2}{4} \ (0 \leq x \leq 2), \ F(x)=1 (x>2){}= 1 \ (x > 2).
  1. (a)
    Find the probability density function of XX.
    (2)
  2. (b)
    Find the exact median and upper quartile of XX.
    (3)
  3. (c)
    Find P(X>1)\mathrm{P}(X > 1).
    (1)
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Part (a)2 marks

  1. f(x)=F′(x)=x2f(x) = F'(x) = \frac{x}{2}
  2. 0≤x≤2; 0 otherwise0 \leq x \leq 2; \ 0 \text{ otherwise}

Part (b)3 marks

  1. F(m)F(m)=12{}= \tfrac{1}{2}⇒m2{}\Rightarrow m^2=2{}= 2
  2. m=2m = \sqrt{2}
  3. F(Q₃) = ¾.Q3=3Q_3 = \sqrt{3}

Part (c)1 mark

  1. 1−F(1)=341 - F(1) = \frac{3}{4}

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