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© Boardworks Ltd of 61 © Boardworks Ltd of 61 A2-Level Maths: Statistics 2 for Edexcel S2.2 Continuous random variables This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation.

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© Boardworks Ltd of 61 Contents © Boardworks Ltd of 61 Relative frequency histograms Probability density functions Mode Cumulative distribution functions Median and quartiles Expectation Variance

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© Boardworks Ltd of 61 A histogram has the important property that the area of each bar is in proportion to the corresponding frequency. A histogram with unequal widths can be drawn by plotting the frequency density on the vertical axis, where: Relative frequency histograms A relative frequency histogram is one in which the area of a bar corresponds to the proportion of the data falling into the corresponding interval. The relative frequency is defined as:

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© Boardworks Ltd of 61 Example: The relative frequency histogram below shows the distribution of ages of the UK population. Relative frequency histograms The area of each bar corresponds to the proportion of the population with ages in that interval. The total area of all the bars is 1.

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© Boardworks Ltd of 61 The distribution of the ages can be modelled by a curve. Relative frequency histograms This curve is called a probability density function.

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© Boardworks Ltd of 61 Contents © Boardworks Ltd of 61 Probability density functions Relative frequency histograms Probability density functions Mode Cumulative distribution functions Median and quartiles Expectation Variance

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© Boardworks Ltd of 61 A probability density function (or p.d.f.) is a curve that models the shape of the distribution corresponding to a continuous random variable. Probability density functions A probability density function has several important properties.

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© Boardworks Ltd of 61 If f(x) is the p.d.f corresponding to a continuous random variable X and if f(x) is defined for a ≤ x ≤ b then the following properties must hold: i.e. the total area under a p.d.f. is 1. Probability density functions 1.

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© Boardworks Ltd of 61 If f(x) is the p.d.f. corresponding to a continuous random variable X and if f(x) is defined for a ≤ x ≤ b then the following properties must hold: 2. Probability density functions i.e. the graph of the p.d.f. never dips below the x -axis.

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© Boardworks Ltd of 61 If f(x) is the p.d.f. corresponding to a continuous random variable X and if f(x) is defined for a ≤ x ≤ b then the following properties must hold: 3. Probability density functions i.e. probabilities correspond to areas under the curve.

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© Boardworks Ltd of 61 Example: Sketch the graph of each of the following functions. Decide in each case whether it could be the equation of a probability density function: Probability density functions

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© Boardworks Ltd of 61 Probability density functions The function is non-negative everywhere. If f represents a p.d.f., then So f(x) could represent a probability density function. 1.

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© Boardworks Ltd of 61 The function is non-negative everywhere. For f to represent a p.d.f. we need to check that But: Probability density functions So f(x) could not represent a probability density function. 2.

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© Boardworks Ltd of 61 Probability density functions The function is clearly negative for some values of x. Consequently f(x) cannot represent a probability density function. 3.

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© Boardworks Ltd of 61 Probability density functions

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© Boardworks Ltd of 61 Question 1: A continuous random variable X is defined by the probability density function Question 1 a)Sketch the probability density function. b)Find the value of the constant k. c)Find P(1 ≤ X ≤ 3).

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© Boardworks Ltd of 61 a) The diagram shows the probability density function. Question 1 Therefore, 12.5 k = 1 b) To find k, we can use the property that

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© Boardworks Ltd of 61 c) P(1 ≤ X ≤ 3) Question 1 = 0.48

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© Boardworks Ltd of 61 Examination-style question 1: A continuous random variable X is defined by the probability density function Examination-style question 1 a)Sketch the probability density function. b)Find the value of the constant k. c)Find P( X > 2).

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© Boardworks Ltd of 61 Examination-style question 1 a)

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© Boardworks Ltd of 61 b) To find k, we can use the property that Note that (5 – x )( x – 2) = 7 x – 10 – x 2 Examination-style question 1 Therefore So,

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© Boardworks Ltd of 61 c) P (X > 2 ) = Examination-style question 1 An alternative method would be to utilise P( X > 2) = 1 – P( X ≤ 2)

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© Boardworks Ltd of 61 Examination-style question 2: The life, T hours, of an electrical component is modelled by the probability density function a) Sketch the probability density function. b) Find the value of the constant k. c) Find P(1500 ≤ T ≤ 2000). Examination-style question 2

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© Boardworks Ltd of 61 Examination-style question 2 Solution: a)

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© Boardworks Ltd of 61 b) To find k, we use the fact that So: Examination-style question 2 Therefore

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© Boardworks Ltd of 61 c) P(1500 ≤ T ≤ 2000) = Examination-style question 2 = (3 s.f.)

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© Boardworks Ltd of 61 Contents © Boardworks Ltd of 61 Mode Relative frequency histograms Probability density functions Mode Cumulative distribution functions Median and quartiles Expectation Variance

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© Boardworks Ltd of 61 Suppose that a random variable X is defined by the probability density function f(x) for a ≤ x ≤ b. The mode of X is the value of x that produces the largest value for f(x) in the interval a ≤ x ≤ b. A sketch of the probability density function can be very helpful when determining the mode. Mode Differentiation could be used to find the mode here.

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© Boardworks Ltd of 61 Example: A random variable X has p.d.f. f(x), where Find the mode. Sketch of f(x): Mode

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© Boardworks Ltd of 61 Mode The mode can be found using differentiation: To find a turning point, we solve Factorize: Check that gives the maximum value: So the mode is.

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© Boardworks Ltd of 61 Contents © Boardworks Ltd of 61 Cumulative distribution functions Relative frequency histograms Probability density functions Mode Cumulative distribution functions Median and quartiles Expectation Variance

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© Boardworks Ltd of 61 The cumulative distribution function (c.d.f.) F(x) for a continuous random variable X is defined as F ( x ) = P( X ≤ x ). Therefore, the c.d.f. is found by integrating the p.d.f.. Example: A random variable X has p.d.f. f ( x ), where Find the c.d.f. and find P (X < 1 ). Cumulative distribution functions

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© Boardworks Ltd of 61 Solution: The c.d.f., F(x) is given by: Cumulative distribution functions To find the constant c we can use the fact that P( X ≤ 0) = 0 (because the random variable X is only non- zero between 0 and 2) Therefore F (0) = 0, i.e. c = 0.

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© Boardworks Ltd of 61 Cumulative distribution functions So the c.d.f. is P ( X < 1) = F (1) =

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© Boardworks Ltd of 61 Contents © Boardworks Ltd of 61 Median and quartiles Relative frequency histograms Probability density functions Mode Cumulative distribution functions Median and quartiles Expectation Variance

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© Boardworks Ltd of 61 The median, m, of a random variable X is defined to be the value such that F ( m ) = P( X ≤ m ) = 0.5 where F is the cumulative distribution function of X. Likewise the lower quartile is the solution to the equation F ( x ) = 0.25 and the upper quartile is the solution to F ( x ) = Median and quartiles

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© Boardworks Ltd of 61 Median and quartiles

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© Boardworks Ltd of 61 Example: A random variable X is defined by the cumulative distribution function: a) Calculate and sketch the probability density function. b)Find the median value. c)Work out P(3 ≤ X ≤ 4). Median and quartiles

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© Boardworks Ltd of 61 a) We can get the p.d.f. by differentiating the c.d.f. So the p.d.f. is Sketch of f ( x ): Median and quartiles

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© Boardworks Ltd of 61 The median must be 3.77 (as the p.d.f. is only non-zero for values in the interval [2, 5]). b)The median, m, satisfies F ( m ) = 0.5. Therefore Median and quartiles

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© Boardworks Ltd of 61 Median and quartiles c) P(3 ≤ X ≤ 4) = F (4) – F (3)

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© Boardworks Ltd of 61 Example: A random variable X has p.d.f. f ( x ), where a)Calculate the cumulative distribution function and verify that the lower quartile is at x = 2. b)Work out the median value of X. Median and quartiles 0 otherwise

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© Boardworks Ltd of 61 Median and quartiles We know that P( X ≤ 1) = 0, i.e., that F (1) = 0. We know that P( X ≤ 4) = 1, i.e. that F (4) = 1. So, So a)

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© Boardworks Ltd of 61 Median and quartiles So To verify that the lower quartile is 2, we simply need to check that F (2) = 0.25: Therefore the lower quartile is 2.

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© Boardworks Ltd of 61 b) The median, m, must lie in the interval [2, 4] because F (2) = To find the median we must solve F ( m ) = 0.5: This can be rearranged to give the quadratic equation: Using the quadratic formula, As 5.63 does not lie in the interval [2, 4], the median must be Median and quartiles m = 5.63 or m = 2.37

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© Boardworks Ltd of 61 Contents © Boardworks Ltd of 61 Expectation Relative frequency histograms Probability density functions Mode Cumulative distribution functions Median and quartiles Expectation Variance

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© Boardworks Ltd of 61 If X is a continuous random variable defined by the probability density function f ( x ) over the domain a ≤ x ≤ b, then the mean or expectation of X is given by E[ X ] is the value you would expect to get, on average. This mean value of X is also sometimes denoted μ. [Note: if the p.d.f. is symmetrical, then the expected value of X will be the value corresponding to the line of symmetry]. We can also find the expected value of g ( X ), i.e. any function of X : Expectation

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© Boardworks Ltd of 61 Example: A random variable X is defined by the probability density function Calculate the value of E[ X ] and E[1 /X ]. Expectation

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© Boardworks Ltd of 61 Expectation So,

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© Boardworks Ltd of 61 Contents © Boardworks Ltd of 61 Variance Relative frequency histograms Probability density functions Mode Cumulative distribution functions Median and quartiles Expectation Variance

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© Boardworks Ltd of 61 If X is a continuous random variable defined by the probability density function f ( x ) over the domain a ≤ x ≤ b, then the variance of X is given by or Variance The standard deviation of X is the square root of the variance. The standard deviation is sometimes denoted by the symbol σ.

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© Boardworks Ltd of 61 Example: A continuous random variable Y has a probability density function f ( y ) where Sketch of f ( y ): The p.d.f. is symmetrical. Therefore E[ Y ] = 2. Variance Calculate the value of Var[ Y ].

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© Boardworks Ltd of 61 Variance Therefore Var[ Y ] =

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© Boardworks Ltd of 61 Example: A continuous random variable x has a probability density function f ( x ) where Calculate a)the mean value, μ. b)the standard deviation, σ. Variance

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© Boardworks Ltd of 61 Variance a)

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© Boardworks Ltd of 61 Variance b) Therefore σ = = 1.40 (3 s.f.)

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© Boardworks Ltd of 61 Examination-style question: The mass, X kg, of luggage taken on board an aircraft by a passenger can be modelled by the probability density function a)Sketch the probability density function and find the value of k. b)Verify that the median weight of luggage is about kg. c)Find the mean and the variance of X. Examination-style question

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© Boardworks Ltd of 61 To find k we use Examination-style question a)

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© Boardworks Ltd of 61 To verify that the median is about , we need to check that P( X ≤ ) = 0.5 Examination-style question b) P( X ≤ ) =

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© Boardworks Ltd of 61 Examination-style question c)

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© Boardworks Ltd of 61 Examination-style question c) Therefore,.

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