Yesterday, Jennifer walked several laps around a track. She made Graph 1 to describe the relationship between time and distance.
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Denote by the random variables representing the integer values
, respectively. Then
and
, where
denotes the discrete uniform distribution over the interval
. So
and
have probability mass functions
We want to find , where
is any integer.
We have six possible choices for :
(i) if , then
is an integer when
;
(ii) if , then
is an integer when
;
(iii) if , then
is an integer when
;
(iv) if , then
is an integer when
;
(v) if or
, then
is an integer only when
in both cases.
If the selection of are made independently, then the joint distribution is the product of the marginal distribution, i.e.
That is, there are 48 possible events in the sample space. We counted 12 possible outcomes in which is an integer, so the probability of this happening is
.