࠵?(࠵? ≤ ࠵? ≤ ࠵?) = ࠵?(࠵? = ࠵?) + ࠵?(࠵? = ࠵?) + ࠵?(࠵? = ࠵?) + ࠵?(࠵? = ࠵?)
࠵?(࠵? ≤ ࠵? ≤ ࠵?) = ࠵?. ࠵?࠵? + ࠵?. ࠵?࠵? + ࠵?. ࠵?࠵? + ࠵?. ࠵?࠵?
࠵?(࠵? ≤ ࠵? ≤ ࠵?) = ࠵?. ࠵?࠵?
e) Find the probability that X is between 1 and 4, exclusive.

࠵?(࠵? < ࠵? < ࠵?) = ࠵?(࠵? = ࠵?) + ࠵?(࠵? = ࠵?)
࠵?(࠵? < ࠵? < ࠵?) = ࠵?. ࠵?࠵? + ࠵?. ࠵?࠵?
࠵?(࠵? < ࠵? < ࠵?) = ࠵?. ࠵?࠵?
f) Calculate the mean of X.
࠵?
࠵?
= % ࠵?
࠵?
࠵?(࠵? = ࠵?
࠵?
)
࠵?
࠵?,࠵?
࠵?
࠵?
= ࠵?
࠵?
࠵?(࠵? = ࠵?
࠵?
) + ࠵?
࠵?
࠵?(࠵? = ࠵?
࠵?
) + ࠵?
࠵?
࠵?(࠵? = ࠵?
࠵?
) + ࠵?
࠵?
࠵?(࠵? = ࠵?
࠵?
) + ࠵?
࠵?
࠵?(࠵? = ࠵?
࠵?
) + ࠵?
࠵?
࠵?(࠵? = ࠵?
࠵?
)
࠵?
࠵?
= ࠵?(࠵?. ࠵?࠵?) + ࠵?(࠵?. ࠵?࠵?) + ࠵?(࠵?. ࠵?࠵?) + ࠵?(࠵?. ࠵?࠵?) + ࠵?(࠵?. ࠵?࠵?) + ࠵?(࠵?. ࠵?)
࠵?
࠵?
= ࠵?. ࠵?࠵?
*Question 3:
A table is provided below. Is this a valid discrete probability distribution? Why or
why not?
x
-10
0
10
P(X=x)
0.01
0.50
0.50
No, because 0.01 + 0.50 + 0.50 = 1.01, but the probabilities must add to 1.
*Question 4:
A table is provided below. Is this a valid discrete probability distribution? Why or
why not?
x
100
200
300
P(X=x)
-0.01
0.49
0.51
No, because P(X=100) = -0.01, but each probability must be between 0 and 1.
** Question 5:
Consider the following data set: 0, 1, 0, 2, 2, 3, 1, 4, 4, 5
a) Construct the empirical probability distribution
x
0
1
2
3
4
5
P(X=x)
2/10
2/10
2/10
1/10
2/10
1/10
Decimals are also acceptable (2/10 = 0.20, 1/10 = 0.10, etc.)
b) What is the shape of this distribution?
The sample size is rather small here, so it’s difficult to say for certain. There are only 10
observed values (n=10), but there are six possible values X can take (0 – 5). In 10
observations, we can only see so many of each value. The shape of this probability
distribution must be taken with a grain of salt. If we were to observe more of the process
that generated the data set, we would probably see a very different shape. For now, it
looks like this is either right-skewed or symmetric. To verify this for yourself, try drawing

a histogram. Remember, frequency goes on the vertical axis (so the vertical axis here
would have 2, 2, 2, 1, 2, 1) and the observations or bins go on the horizontal axis (so here
the horizontal axis would have 0, 1, 2, 3, 4, 5).

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- Fall '19
- Probability distribution, Probability theory, Discrete probability distribution, Continuous probability distribution