HW6: Name ______________________________ Class Number _________________ 1. A coin is to be tossed 3 times. Let X count the number of heads tossed. Draw a spinner that has the same distribution as X.The 8 possibilities are:HHHTHHHHT THTHTH TTHHTT TTT2. A game is played as follows: You roll a die. If you roll a 1, 2, or 3 you move ahead 1 space. If you roll a 4 or 5 you move ahead 2 spaces. If you roll a 6 you move ahead 3 spaces. Let X denote the number of spaces that you move. a) Write out the pmf in a chart.b) Draw the pmf (The stick figure shown in the text)c) Make a spinner that has the same distribution as X.1HW7A: Name ______________________________ Class Number _________________ 1. Use the pmf of the random variable X to determine the mean of X.x 1 2 3 4 5 6f (x) .3 .2 .1 .2 .1 .1xf(x) .3 .4 .3 .8 .5 .6( ) 2.9xf xm=S =2. Use the pmf of the random variable X to determine the mean of X.x1 2 3 4 5 6 7 8 9 10( )f x.1 .2 .05 .05 .15 .05 .1 .1 .15 .05.1 .4 .15 .2 .75 .3 .7 .8 1.35 .5( ) 5.25xf xm=S =3. A game is played as follows: You roll a die. If you roll a 1, 2, or 3 you move ahead 1 space. If you roll a 4 or 5 you move ahead 2 spaces. If you roll a 6 you move ahead 3 spaces. Let X denote the number of spaces that you move. Determine the expected value of X.x 1 2 3f(x) 3/6 2/6 1/6xf(x) 3/6 4/6 3/6( ) 10/ 6xf xm=S = 4. A game is played as follows: A die is rolled. If the number of spots is even, you win that value in dollars. If the number of spots is odd, you lose $4. a) Draw a spinner that has the same distribution as the game. b) Determine the expected value of the game.1 1 1 3( ) 2 4 6 4 06 6 6 6xf x� � � � � � � �=S = + + - =� � � � � � � �� � � � � � � �m2HW7C: Name ______________________________ Class Number _________________ 1. Use the pmf to determine the mean of the random variable X.a) x 1 2 3 4 5 6 7 8f (x) .05 .15 .15 .25 .05 .15 .1 .1.05 .3 .45 1.0 .25 .9 .7 .8 4.45m= + + + + + + + =b) x 2 4 6 8 10 12 14 16f (x) .02 .04 .04 .12 .14 .14 .15 .35.04 .16 .24 .96 1.4 1.68 2.1 5.6 12.18m= + + + + + + + =2. For each game below, determine the expected value.a) Three coins are to be tossed. If you toss 0 or 3 heads you win $10. If not, you lose $3.For part a only, also make a spinner with the same distribution as this game.x f(x) xf(x)10 2/8 20/8-3 6/8 -18/820 / 8 18/8 2 / 8m= - =b) Two dice are rolled. If the sum is 2, you win $2. If the sum is 12, you win $3. If the sum is 3, 4, 9, 10, or 11 you win $1. For every other sum you lose $1.x 2 3 1 -1f(x) 1/36 1/36 14/36 20/36xf(x) 2/36 3/36 14/36 -20/36(2 / 36) (3/ 36) (14 / 36) (20 / 36) 1/ 36m= + + - =-c) A single die is rolled. If you roll a 1, 2. or a 3, you lose $5. If you roll a 4 or 5, you win $3. If you roll a 6 you win $8. x f(x) xf(x)-5 3/6 -15/63 2/6 6/68 1/6 8/6( 15/ 6) (6 / 6) (8 / 6) 1/ 6m= - + + =-3HW7D: Name ______________________________ Class Number _________________ Suppose that our company performs DNA analysis for a law enforcement agency. We currently have 2 machinesthat are essential to performing the analysis. When an analysis is performed, the machine is in use for the entire day.Thus, we can perform at most two DNA analyses per day. Based on past experience, the distribution of analysesneeding to be performed on any given day are as follows:Jobs 0 1 2 3 4 5 or moref(x) .08 .12 .21 .24 .21 .14x 0 0 0 900 900 900xf(x) 0 0 0 216 189 126On days with three or more available jobs to perform, since we cannot perform more than two, the law enforcementagency gives the extra jobs to our competitor. We are considering purchasing a third machine. Each day that the machine is in use, we profit $900. What is theyearly expected value of this new machine? (Assume 365 days per year – no weekends or holidays)a) Determine the expected value per day of the third machine for the entire year.Two or fewer analyses yield no additional dollars since we can already handle two. Three analyses yield anadditional analysis that we would not have been able to do with only 2 machines. So we earn an extra $900. If wereceive four orders, we can handle three of them for an additional profit of $900. The same is true for 5 or more.Our daily expected value is then 216 189 126 531+ + =.The expected value for the year is $531 365 $193,815� =. b) Determine the expected value of adding a fourth machine for the entire year.Jobs 0 1 2 3 4 5 or moref(x) .08 .12 .21 .24 .21 .14x 0 0 0 0 900 900xf(x) 0 0 0 0 189 126Daily expected value is $189 $126 $315+ =.Expected value for the year is $315 365 $114,975� =.4HW7E: Name ______________________________ Class Number _________________ Suppose that our company performs DNA analysis for a law enforcement agency. We currently have 1 machine thatis essential to performing the analysis. When an analysis is performed, the machine is in use for half of the day.Thus, each machine of this type can perform at most two DNA analyses per day. Based on past experience, thedistribution of analyses needing to be performed on any given day are as follows:Jobs 0 1 2 3 4 5 6 7 or moref(x) .05 .11 .13 .17 .24 .14 .10 .06x0 0 0 900 1800 1800 1800 1800xf(x)0 0 0 153 432 252 180 108On days with three or more available jobs to perform, since we cannot perform more than two, the law enforcementagency gives the extra jobs to our competitor. We are considering purchasing a second machine. For each analysis that the machine is in use, we profit $900.What is the yearly expected value of this new machine? (Assume 365 days per year – no weekends or holidays)a) Determine the expected value per day of the second machine and for the entire year.Daily expected value is $1125 and the yearly is $410,625.b) Determine the expected value of adding a third machine for the entire year.Jobs 0 1 2 3 4 5 6 7 or moref(x) .05 .11 .13 .17 .24 .14 .10 .06x0 0 0 0 0 900 1800 1800xf(x)0 0 0 0 0 126 180 108Daily …