Probability density function TYS 1

Properties.	1. The following diagram shows a probability density function as a line: (i) Find the equation for the probability density function. (ii) Check that it is a probability density function. Answer. (i) P(x) = 0.4 - 0.08x [0, 5]. The function here is just the usual equation for the line shown. (ii) The area below the line: = 0.5 × 0.4 × 5 = 1.0. Hence a pdf.
	2. The following diagram shows a possible probability density function as two lines: (i) Find the possible probability density function (it consists of 2 parts). (ii) Check if it is a probability density function. Answer.(i) P(x) = 0.5 - 0.2x [0,1] and P(x) = 0.45 - 0.15x [1, 3]. The function combines the equations for the two lines shown. (ii) Area Trap = 0.4 Area Tri = 0.3 Sum = 0.7.
	3. The following diagram shows a probability density function as two lines: (i) Find the probability density function (it consists of 2 parts). (ii) Check that it is a probability density function. Answer. P(x) = 0.4 - 0.4x/3 [0,3] and P(x) = 0.2x - 0.6 [3, 5]. The function combines the equations for the two lines shown.
	4. Show that the function does not satisfy either of the properties for a probability density function.
	5. (i) Show that when a = 1/6. (ii) Draw the sketch of y = a( x² - 1) for that value of a and in the domain [0, 3] showing the x and y intercepts. (iii) Explain why the function y = a(x² - 1) is not a probability function.
	6. A probability density function is defined as in the domain [0, 3]. Find the value of a. Answer. a= 2/9.
	7. A function is defined as (i) Sketch this function and find the function value at x = 2. (ii) Show that, for the domain [0, 2], this function cannot be regarded as a probability density function. (iii) If the function is a probability density function, find the value of a and hence write down the probability density funtion for the domain [0, 2]. (iv) Sketch the probability density function and find Pr (x = 2). Answer. P(x = 2) = 0.5.
	8. The probability density function for the random variable x is defined in the domain [0, 2] as . (i) Find the value of t (to 4 significant figures). (ii) Draw a sketch of the probability distribution. (iii) Hence find the probability that x < 1. (2 decimal places).
Uniformly distributed functions.	9. David has recently taken up kayaking on a nearby lake to give himself exercise during lockdown. He aims to be on the water for 30 minutes but is presently averaging 24 minutes. The time he spends on the water varies between 18 and 30 minutes. (i) Explain why the distribution of kayaking time can be regarded as being a uniform distribution. (ii) What is the probability that David will spend less than 20 minutes on the water? (iii) What is the probability that David will spend between 20 and 24 minutes on the water? (iv) What is the probability that David will spend at least 29 minutes on the water? Answer.(i) Time is continuous and David can stop at any time with equal probability. (ii) P(<20) = 2/6 = 0.33. (iii) P(20(iv) P(>29) = 1/6 = 0.17.
	10. The International Space Station (ISS) completes an orbit of Earth every 90 minutes (say - the time depends on its height and the time can be up to about 93 minutes). Hence the probability of seeing the Space Station in one minute from now is 1/90 (without checking on the app ISS Live Now). This probability distribution can therefore be regarded as being uniformly distributed. (i) Draw a graph of the probability of seeing the Space Station P(sighting) against time (in minutes) for the domain [0, 90]. (ii) Write down the probability density function for sighting the ISS. (ii) What is the probability of going outside now and seeing the ISS within 5 minutes? (iii) What is the probability of seeing the ISS pass between 30 and 40 minutes from now? (iv) What is the probability of having to wait for at least 60 minutes before seeing the ISS pass?
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Expected value and variance.	14. Using the strategy outlined elsewhere, prove that, for a uniform probability distribution function, the variance can be expressed as . You can assume that the factorisation of b³ - a³ = (b - a)(b² - ab + b²)
	15. The random variable X has a probability density function defined as: Find, correct to two decimal places: (i) P (3.1 ≤ x ≤ 4.5). (ii) E (X). (iii) Var (X). Answer.(i) P (3.1 ≤ x ≤ 4.5) = 4.3 (ii) E(X) = 3.33. (iii) Var (X) = 1.61.
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	17. In relation to the uniform distribution for the ISS in Q 7 above, calculate the expected mean time for seeing the ISS pass overhead.

Find the mode.