Download MAT 332, Probability

MAT 470, Expected Value and Variance
Name:_____________________
1. For the following probability distribution,
a). determine the missing probability value
–5
0.20
y
p(y)
–2
0.35
3
_______?
6
0.15
b). Determine the expected value, E(Y) =
c). Compute E( Y2 ) =
d). Determine the variance, V( Y ) =
2. Six balls, numbered 1, 2, 3, 4, 5, and 6, are placed in an urn. A pair of balls is randomly
selected from the six and their numbers noted. Let Y be the absolute value of the difference
of the two sampled numbers.
a). Give the sample space for this experiment. Note the outcome {1, 5} is the same as {5, 1}.
b). Construct the probability distribution for Y.
y
p(y)
c). Determine the expected value, E(Y) =
d). Determine the variance, V( Y ) =
3. A drink is selected from a cooler containing twenty 12-ounce cans, thirty-five 20-ounce
bottles, and twenty-five 24-ounce bottles. Let X be the size (in ounces) of the drink selected.
a). Complete the probability distribution:
x
p(x)
12
20
24
b). determine the expected value for the size of the drink selected.
c). determine the variance for the size of the drink selected.
4. Suppose a game uses a spinner to determine Y, the number of
places you may move your playing piece. Suppose the spinner tends to
stop on “3” and “6” twice as often as it stops on the other numbers.
a). Construct the probability distribution for Y.
2
1
3
6
y
1
2
3
4
5
6
p(y)
b). Determine the mean, E(Y) =
c). Determine the variance, V( Y ) =
5. Suppose the density function for a continuous random variable Y is given by
y<0
 0,
 2y

,
0 y3
f ( y )   15
1  0.2 y, 3  y  5

5 y
 0,
a). Determine the probability, P( Y < 2 ) =
b). Determine the probability, P( Y < 4 ) =
5
4
k y (3  y ), 0  y  3
6. Suppose that Y has density function f ( y )  
elsewhere
 0,
a). Use the fact that



f ( y)dy  1 to find the value of k that makes f ( y )
a valid probability density function.
b). Determine the probability, P( Y < 2 ).
7. Define a continuous random variable Y with distribution function given by
y0
 0,
F ( y)  
 0.2 y
,y0
1  e
a). Determine the density function for Y.

f ( y )  F ( y )  

,y0
,y0
b). Determine the expected value, E(Y)
c). Determine the probability P( 4 < Y < 8 ) =
d). Determine the probability P( Y > 2 ) =
e). Determine the probability P( Y > 6 | Y > 2 ) =