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4CCM141A/5CCM141B – Probability and Statistics I Exercise Sheet 6 1. The median of a continuous random variable having a Cumulative Distribution Function (CDF) F ( y ) is the specific value y = m such that F ( m) = 12 . That is, a random variable is just as likely to be larger than its median as it is to be smaller. Find the median of Y if Y is a. uniformly distributed over [ a, b] , i.e. Y ~ Uniform [a, b] . ( ) b. normal with parameters µ ,σ , i.e. Y ~ N µ ,σ 2 . c. exponential with mean θ , i.e. Y ~ exponential (θ ) . d. The mode of a continuous random variable having a Probability Density Function (PDF) p ( y ) is the specific value of y for which p ( y ) attains its maximum. (i) Compute the mode of Y in the cases a,b,c above. (ii) What is the value of the density on the mode? Comment on the values you get. 2. A stick of length 1 is split at a point U that is uniformly distributed over ( 0,1) . a. Determine the expected length of the piece that contains the point p, 0 ≤ p ≤ 1. b. What is the value of p that maximizes the expected length? What is the corresponding maximal length? Hint: you might want to define the following function of the RV U : Lp (U ) = the length of the substick that contains p . Then find the expectation value of this variable Lp (U ) . 3. (Textbook 4.58+4.59) a. Use the relevant table to find the following probabilities for a standard normal random variable Z : (i) P ( 0 ≤ Z ≤ 1.2 ) (ii) P ( −.9 ≤ Z ≤ 0 ) (iii) P (.3 ≤ Z ≤ 1.56 ) (iv) P ( −.2 ≤ Z ≤ .2) b. If Z is a standard normal random variable, find the value z0 such that (i) P ( Z > z0 ) = .5 (ii) P ( Z < z0 ) = .8643 (iii) P ( − z0 < Z < z0 ) = .90 (iv) P ( − z0 < Z < z0 ) = .99 4. (Textbook 4.71) Wires manufactured for use in a computer system are specified to have resistances between .12 and .14 ohms. The actual measured resistances of the wires produced by company A have a normal probability distribution with mean .13 ohm and standard deviation .005 ohm. a. What is the probability that a randomly selected wire from company A’s production will meet the specifications? b. If four of these wires are used in each computer system and all are selected from company A, what is the probability that all four in a randomly selected system will meet the specifications? 5. (Textbook 4.75+4.76) a. A soft-drink machine can be regulated so that it discharges an average of µ ounces per cup. If the ounces of fill are normally distributed with standard deviation 0.3 ounce, give the setting for µ so that 8-ounce cups will overflow only 1% of the time. b. This machine has standard deviation σ that can be fixed at certain levels by carefully adjusting the machine. What is the largest value of σ that will allow the actual amount dispensed to fall within 1 ounce of the mean with probability at least .95? 6. (Textbook 4.80) Assume that Y is Normally distributed with mean µ and standard ( ) deviation σ (i.e. Y ~ N µ ,σ 2 ). After observing a value Y , a mathematician constructs a rectangle with length L = Y and width W = 3 Y . Let A denote the area of the resulting rectangle. What is the expected area A ? 7. (Textbook 4.92) The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula C = 100 + 40Y + 3Y 2 relates the cost C of completing this operation to the square of the time to completion. Find the mean and variance of C. Chapter 4 Continuous Variables and Their Probability Distributions 8. Textbook 4.88, 4.90 The magnitude of earthquakes recorded in a region of North America can be modelled as an exponential distributionrandom with mean 2.4,with as measured on the Richter scale. *4.191 having Suppose that Y is a continuous variable distribution function given by Find F(y) and f (y). We often interested theprobability probabilitydensity that anfunction earthquake striking this are region will in conditional probabilities of the form P(Y ≤ y|Y ≥ c) for a constant c. a. exceed 3.0 on the Richter scale. a Show that, for y ≥ c, b. fall between 2.0 and 3.0 on the Richter scale. F(y) − F(c) P(Y this ≤ y|Y ≥ c)what = is the probability . c. Of the next 10 earthquakes to strike region, that at least one 1 − F(c) will 5.0 on Richterinscale? b exceed Show that thethe function part (a) has all the properties of a distribution function. c If the length of life Y for a battery has a Weibull distribution with m = 2 and α = 3 (with measurements in years), find the probability that the battery will last less than four years, 9. Textbook 4.192 (You can use the fact that Γ(1 / 2) = π ) given that it is now two years old. *4.192 The velocities of gas particles can be modeled by the Maxwell distribution, whose probability density function is given by ! m "3/2 2 v 2 e−v (m/[2K T ]) , v > 0, f (v) = 4π 2π K T where m is the mass of the particle, K is Boltzmann’s constant, and T is the absolute temperature. a b *4.193 Find the mean velocity of these particles. The kinetic energy of a particle is given by (1/2)mV 2 . Find the mean kinetic energy for a particle. Because F(y) − F(c) 1 − F(c) Optional exercises (from Textbook): 4.18, 4.19+4.25, 4.48, 4.74, 4.91. has the properties of a distribution function, its derivative will have4.51+4.52, the properties of4.80, a probability density function. This derivative is given by P(Y ≤ y|Y ≥ c) = f (y) , y ≥ c. 1 − F(c) We can thus find the expected value of Y , given that Y is greater than c, by using # ∞ 1 E(Y |Y ≥ c) = y f (y) dy. 1 − F(c) c If Y , the length of life of an electronic component, has an exponential distribution with mean 100 hours, find the expected value of Y , given that this component already has been in use for 50 hours. *4.194 We can show that the normal density function integrates to unity by showing that, if u > 0, # ∞ 1 1 2 e−(1/2)uy dy = √ . √ u 2π −∞ This, in turn, can be shown by considering the product of two such integrals: $# ∞ % $# ∞ % # ∞# ∞ 1 1 2 2 −(1/2)uy 2 −(1/2)ux 2 e dy e dx = e−(1/2)u(x +y ) d x d y. 2π 2π −∞ −∞ −∞ −∞ By transforming to polar coordinates, show that the preceding double integral is equal to 1/u.