Consider the experiment of simultaneously tossing a die and a coin. Let X denote the number of heads and Y denote the number of spots showing on the die.
a)construct a two-way table for the joint pdf
b)Let Z=X+Y . Use the joint pdf to find the distribution or Z.
To construct a two-way table for the joint probability density function (pdf) of X and Y, we need to consider all possible outcomes of tossing a die and a coin simultaneously.
a) Constructing a Two-Way Table for the Joint pdf:
Let's break down the possibilities for both X (number of heads on the coin) and Y (number of spots on the die).
For X, there are two possible outcomes: 0 heads (Tails) or 1 head (Heads).
For Y, there are six possible outcomes: 1, 2, 3, 4, 5, or 6 spots on the die.
Now, we can create a two-way table for the joint pdf:
```
| Y=1 | Y=2 | Y=3 | Y=4 | Y=5 | Y=6 |
------------------------------------------------------------------
X=0 | P(X=0,Y=1)| P(X=0,Y=2)| P(X=0,Y=3)| P(X=0,Y=4)| P(X=0,Y=5)| P(X=0,Y=6)|
------------------------------------------------------------------
X=1 | P(X=1,Y=1)| P(X=1,Y=2)| P(X=1,Y=3)| P(X=1,Y=4)| P(X=1,Y=5)| P(X=1,Y=6)|
```
To find the probabilities in each cell, we need to consider that the outcomes of the die and the coin are independent. Since the die has a uniform distribution, each outcome (1-6) is equally likely with a probability of 1/6.
Therefore, we can calculate the joint probabilities by multiplying the probability of each individual outcome. For example, P(X=0, Y=1) would be P(X=0) * P(Y=1) = (1/2) * (1/6) = 1/12.
b) Finding the Distribution of Z:
Since Z = X + Y, we can use the joint pdf to find the distribution of Z by summing up the probabilities of all the cells in the two-way table with the same Z value.
Let's calculate the distribution for Z:
```
Z | Probability
------------------------
2 | P(X=0,Y=2) + P(X=1,Y=1)
3 | P(X=0,Y=3) + P(X=1,Y=2)
4 | P(X=0,Y=4) + P(X=1,Y=3)
5 | P(X=0,Y=5) + P(X=1,Y=4)
6 | P(X=0,Y=6) + P(X=1,Y=5)
7 | P(X=1,Y=6)
```
Simply add up the probabilities for each Z value to obtain the distribution of Z.
Note: The joint pdf and the distribution of Z can also be visualized using a probability mass function (pmf) or a probability density function (pdf) plot.