Let X be a random variable that records the outcome of a dice toss. The probability distribution for X is as follows:
Outcome: 1, 2, 3, 4, 5, 6
Probability: 1/6, 1/6, 1/6, 1/6, 1/6, 1/6
Suppose now that we toss two dice. X1 records the outcome of the first toss, and X2 records the outcome of the second toss. We are interested in the sum of the two dice tosses,
Y= X1+ X2.
Do you think that X1and X2 are independent random variables?
Yes, X1 and X2 are independent random variables. The outcome of one dice toss does not affect the outcome of the other dice toss. Each toss is independent and has the same probability distribution, making X1 and X2 independent.
To determine if X1 and X2 are independent random variables, we need to compare the joint probability distribution of X1 and X2 to the product of their individual probability distributions. If the joint probability distribution is equal to the product of the individual probability distributions, then X1 and X2 are independent.
In this case, X1 and X2 are the outcomes of two separate dice tosses, so we can calculate the joint probability distribution by considering all possible outcomes of X1 and X2. Since each dice has 6 possible outcomes (numbers 1 to 6), there are a total of 6 * 6 = 36 possible outcomes for X1 and X2 combined.
To calculate the joint probability distribution, we need to find the probability of each possible outcome. We can do this by multiplying the probability of X1 with the probability of X2 for each outcome.
For example, when X1 = 1 and X2 = 1, the joint probability is the product of the probability of X1 = 1 (1/6) and the probability of X2 = 1 (also 1/6), which is (1/6) * (1/6) = 1/36.
By repeating this process for all possible outcomes, we can create the joint probability distribution table:
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| X1/X2 | 1 | 2 | 3 | 4 | 5 | 6 |
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| 1 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |
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| 2 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |
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| 3 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |
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| 4 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |
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| 5 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |
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| 6 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |
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Now, let's compare the joint probability distribution to the product of the individual probability distributions. If they are equal, then X1 and X2 are independent random variables.
Multiplying the individual probability distributions, we get:
P(X1 = 1) * P(X2 = 1) = (1/6) * (1/6) = 1/36
Comparing this product to the joint probability distribution, we can see that they are equal for all possible outcomes. Therefore, X1 and X2 are independent random variables.
In conclusion, X1 and X2 are independent random variables in this scenario.
To determine whether X1 and X2 are independent random variables, we need to check if the joint probability distribution of X1 and X2 can be expressed as the product of the marginal probability distributions of X1 and X2.
Let's calculate the joint probability distribution of X1 and X2 by considering all possible outcomes of the two dice tosses:
Outcome of X1, X2: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
Probability: 1/36, 1/36, 1/36, 1/36, 1/36, 1/36
Outcome of X1, X2: (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
Probability: 1/36, 1/36, 1/36, 1/36, 1/36, 1/36
Outcome of X1, X2: (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
Probability: 1/36, 1/36, 1/36, 1/36, 1/36, 1/36
Outcome of X1, X2: (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
Probability: 1/36, 1/36, 1/36, 1/36, 1/36, 1/36
Outcome of X1, X2: (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
Probability: 1/36, 1/36, 1/36, 1/36, 1/36, 1/36
Outcome of X1, X2: (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
Probability: 1/36, 1/36, 1/36, 1/36, 1/36, 1/36
Now, let's compare the joint probability distribution with the product of the marginal probability distributions:
For example, the joint probability of (1, 1) is 1/36.
The product of the marginal probabilities P(X1 = 1) * P(X2 = 1) is (1/6) * (1/6) = 1/36.
As we can see, the joint probability and the product of marginal probabilities are the same for all possible outcomes. Therefore, we can conclude that X1 and X2 are independent random variables.