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a3pr12f06

# a3pr12f06 - Problems and results for the twelfth week...

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Problems and results for the twelfth week Mathematics A3 for Civil Engineering students 1. Which of the following functions can be a distribution function? (a) F ( x ) = braceleftBigg 1 + e 1 x , if x > 1 , 0 , otherwise (b) F ( x ) = 2 2 x + 1 , if x 0 , 0 , otherwise (c) F ( x ) = braceleftBigg 1 e x , if x 0 , 0 , otherwise (d) F ( x ) = 0 , if x 0 , x 4 · (4 x ) , if 0 < x 2 , 1 , if x > 2 2. Which of the following functions can be a probability density function? (a) f ( x ) = 2 x , if x > 1 , 0 , otherwise (b) f ( x ) = sin( x ) 2 , if 0 < x < 2 , 0 , otherwise (c) f ( x ) = 3 x 1 ln(3) , if x 0 , 1 3 sin parenleftBig x 2 parenrightBig , if 0 < x < π, 0 , otherwise (d) f ( x ) = braceleftBigg 2e 2 x , if x > 0 , 0 , otherwise 3. Compute the expectation and variance of a random variable X with density f ( x ) = braceleftBigg 2 x , if 0 < x < 1 , 0 , otherwise. 4. Find the probabilities P { m σ < X < m + σ } and P { m 2 σ < X < m +2 σ } in the previous question, where m stands for expectation and σ 2 for variance. 5. Consider the function f ( x ) = c (2 x x 3 ) , if 0 < x < 5 2 , 0 , otherwise. Could f be a probability density function? If so, determine c . 1

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Repeat if f ( x ) were given by f ( x ) = c (2 x x 2 ) , if 0 < x < 5 2 , 0 , otherwise. 6. A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of liters is a random variable with probability density function f ( x ) = braceleftBigg 5(1 x ) 4 , if 0 < x < 1 , 0 , otherwise, what needs the capacity of the tank be so that the probability of the supply’s being exhausted in a given week is 0.01? 7. Compute E ( X ) if X has a density function given by (a) f ( x ) = 1 4 x e x/ 2 , if x > 0 , 0 , otherwise; (b) f ( x ) = braceleftBigg c (1 x 2 ) , if 1 < x < 1 , 0 , otherwise; (c) f ( x ) = 5 x 2 , if x > 5 , 0 , otherwise? 8. The lifetime, in days, of a part of a machine is a random variable with density f ( x ) = 2 /x 3 when x > 1. What is the probability that this part still works on February 1, if we bought it on January 26? Should we rather buy the part with density function f ( x ) = 1 /x 2 when x > 1? What is the average lifetime for the two types of parts? 9. What is the probability that, if I wake with a start in the middle of the night, the minute hand of the clock is on the right hand-side of the vertical line that passes through the center of the clock? And what is the probability that the minute hand points inside the 1 12 part of the circle that is located between the numbers 5 and 6? 10. What is the probability that, out of three independent points chosen in (0 , 1), precisely one falls in each of the intervals (0 , 1 3 ), ( 1 3 , 2 3 ), ( 2 3 , 1)? 11. A long and high fence consists of columns of diameter D , placed in a distance of L from each other. I throw a ball of diameter d from a long distance, with my eyes closed to this fence. The ball either hits one of the columns, or it passes between them without touching. What is the probability that the ball passes without touching?
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