As in the previous lecture clip, consider the same example of two rolls of a 4-sided die, with all 16 outcomes equally likely. As before, let X be the result of the first roll and Y be the result of the second roll. Define W=XY . Find the numerical values of pW(4) and pW(5) .
1. 3/16
2. 0
To find the numerical values of pW(4) and pW(5), we need to determine the possible outcomes that result in the value 4 or 5 for the random variable W.
The values of X and Y can range from 1 to 4, as they represent the possible outcomes of rolling a 4-sided die.
To calculate pW(4), we need to find all the possible combinations of X and Y that result in a product of 4.
The possible combinations are:
(1, 4)
(2, 2)
(4, 1)
Therefore, pW(4) = P(XY = 4) = P((1, 4) or (2, 2) or (4, 1)).
Since all 16 outcomes are equally likely, the probability of each outcome is 1/16. So, pW(4) = 3/16.
To calculate pW(5), we need to find all the possible combinations of X and Y that result in a product of 5.
The possible combinations are:
(1, 5)
(5, 1)
Therefore, pW(5) = P(XY = 5) = P((1, 5) or (5, 1)).
Since all 16 outcomes are equally likely, the probability of each outcome is 1/16. So, pW(5) = 2/16 = 1/8.
Therefore, the numerical values of pW(4) and pW(5) are 3/16 and 1/8, respectively.
To find the numerical values of pW(4) and pW(5), we need to determine the probability distribution function (PDF) of the random variable W.
Step 1: Determine the possible values of W.
Since W = XY, where X and Y represent the results of the first and second rolls of the die, respectively, we can deduce the possible values of W by considering all possible outcomes of X and Y.
The possible values of X range from 1 to 4, and similarly, the possible values of Y range from 1 to 4. Therefore, the possible values of W are the products of all possible pairs of X and Y, which are:
W = {1, 2, 3, 4}.
Step 2: Calculate the probabilities for each value of W.
Since all 16 outcomes are equally likely, we can calculate the probabilities by counting the number of outcomes that result in each value of W and dividing it by the total number of possible outcomes (16).
For W = 1:
The possible pairs of X and Y that give W = 1 are: (1, 1).
Therefore, pW(1) = Number of outcomes with W = 1 / Total number of possible outcomes
= 1 / 16
For W = 2:
The possible pairs of X and Y that give W = 2 are: (1, 2) and (2, 1).
Therefore, pW(2) = Number of outcomes with W = 2 / Total number of possible outcomes
= 2 / 16 = 1 / 8
Similarly, we can calculate pW(3), pW(4), pW(5), etc., by counting the number of outcomes for each value of W.
For W = 3:
The possible pairs of X and Y that give W = 3 are: (1, 3), (3, 1), (2, 2).
Therefore, pW(3) = Number of outcomes with W = 3 / Total number of possible outcomes
= 3 / 16
For W = 4:
The possible pairs of X and Y that give W = 4 are: (1, 4), (4, 1), (2, 2).
Therefore, pW(4) = Number of outcomes with W = 4 / Total number of possible outcomes
= 3 / 16
For W = 5:
The possible pairs of X and Y that give W = 5 are: (2, 3), (3, 2).
Therefore, pW(5) = Number of outcomes with W = 5 / Total number of possible outcomes
= 2 / 16 = 1 / 8
Step 3: Calculate the numerical values of pW(4) and pW(5) using the results obtained.
From our calculations:
pW(4) = 3 / 16 ≈ 0.1875
pW(5) = 1 / 8 = 0.125
Therefore, the numerical values of pW(4) and pW(5) are approximately 0.1875 and 0.125, respectively.
1. 3/16
2. 1/16