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MAT 470, Worksheet 5 Name:_____________________ 1. One of the continuous random variables that we have not discussed is the Laplace random variable Y whose moment generating function is given by a 2 ebt m(t ) 2 2 , where a and b are constants with a > 0. a t Use this moment generating function to help you determine the mean of Y in terms of the constants, a and/or b. 2. A continuous random variable Y is uniformly distributed on the interval [2, 8]. Hence the density function is given by 1/ 6, 2 y 8; f ( y) 0, otherwise. a). Determine the probability, P( Y < 6 ) = b). Determine the cumulative distribution function F(y). , y 2; F ( y) , 2 y 8; , y 8. c). Determine the 4th moment of Y, E( Y4 ) = d). Define a random variable W = 3Y + 2. Determine the mean and variance of W. 3. Consider the random variables whose familiar moment generating functions are given below. e 4 t et a). For a random variable Y, suppose mY (t ) . 3t i). Determine E(Y) and V(Y). ii). Determine the expected value E (eY / 2 ) b). For a random variable X, suppose mX (t ) 5 . 5 12t i). Determine E(Y) and V(Y). ii). Define W = 3X. Give the moment generating function for W, mW (t ) . 4. Suppose the height Y (in inches) of a crop of wheat is normally distributed with standard deviation 2.5 inches. a). Determine the average height of the hay if 67% of the crop measures less than 31 inches in height. b). Now, suppose the average height is 30 inches and standard deviation is 2.5 inches. Compute the following probabilities. i). P( 32 < Y < 35 ) = ii). P( Y > 35 ) = 5. The amount of fill dispensed by a bottling machine is normally distributed with standard deviation = 0.6 ounces. a). If the dispenser fills 16 ounce cups such that the cups overflow with probability 0.005, what is the mean for the amount dispensed by the machine? b). If the machine is adjusted so the amount dispensed is normally distributed with = 16 and = 0.6 ounces, what is the probability that a 17-ounce cup is filled to overflowing (i.e., more than 17 ounces is dispensed)? 6. Define a continuous random variable Y with distribution function given by y0 0, F ( y) 0.2 y ,y0 1 e a). Determine the density function for Y. f ( y) ,y0 ,y0 b). Give the mean, E(Y) c). Determine the probability P( 4 < Y < 8 ) = d). Determine the conditional probability P( Y > 8 | Y > 6 ) = 7. Consider a continuous random variable Y whose moment generating function is 3e 3t given by m(t ) . Use this moment generating function to help you determine the 3t mean, E(Y). 8. Let Y be an exponentially distributed continuous random variable with moment generating function m(t ) 2 1 . 2 3 t 1 32 t a). Give the mean and variance of Y. b). Recall the moments of Y are given by E( Yk ) = (k!)k . For W = Y2 + 3 , determine the second moment of W. E( W2 ) = E( Y4 ) +6E( Y2 ) + 9 =