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

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Unformatted text preview: 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 , , otherwise (b) F ( x ) = 2 − 2 x + 1 , if x ≥ , , otherwise (c) F ( x ) = braceleftBigg 1 − e − x , if x ≥ , , otherwise (d) F ( x ) = , if x ≤ , 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 , , otherwise (b) f ( x ) = sin( x ) 2 , if 0 < x < 2 , , otherwise (c) f ( x ) = 3 x − 1 ln(3) , if x ≤ , 1 3 sin parenleftBig x 2 parenrightBig , if 0 < x < π, , otherwise (d) f ( x ) = braceleftBigg 2e − 2 x , if x > , , otherwise 3. Compute the expectation and variance of a random variable X with density f ( x ) = braceleftBigg 2 x , if 0 < x < 1 , , 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 , , otherwise. Could f be a probability density function? If so, determine c . 1 Repeat if f ( x ) were given by f ( x ) = c (2 x − x 2 ) , if 0 < x < 5 2 , , 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 , , 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 > , , otherwise; (b) f ( x ) = braceleftBigg c (1 − x 2 ) , if − 1 < x < 1 , , otherwise; (c) f ( x ) = 5 x 2 , if x > 5 , , 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?...
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a3pr12f06 - Problems and results for the twelfth week...

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