Question

a coin is tossed 3 times. let x denotes the number of heads and y denotes the absolute difference between the number of heads and tails. find the joint probability mass function and the marginal densities for x and y .

Answer 1

The joint PMF: P(X = 0, Y = 3) = P(X = 3, Y = 3) =
`(1)/(8)\)`. Marginal PMFs for X and Y match joint values.

Let's define the random variables:

X = Number of heads in 3 tosses of a coin

Y = Absolute difference between the number of heads and tails

To find the joint probability mass function (PMF), we'll first list all possible outcomes when tossing a coin 3 times:

For X, the number of heads can be 0, 1, 2, or 3.

For Y, the absolute difference between heads and tails can be 0, 1, 2, or 3.

When X = 0, Y = |0 - 3| = 3

When X = 1, Y = |1 - 2| = 1

When X = 2, Y = |2 - 1| = 1

When X = 3, Y = |3 - 0| = 3

Now, let's calculate the probabilities for each outcome:

P(X = 0) = Probability of getting 0 heads in 3 tosses = 1/8

P(X = 1) = Probability of getting 1 head in 3 tosses = 3/8

P(X = 2) = Probability of getting 2 heads in 3 tosses = 3/8

P(X = 3) = Probability of getting 3 heads in 3 tosses = 1/8

Now, let's find the probabilities for Y corresponding to each X value:

When X = 0, Y = 3, so P(Y = 3 | X = 0) = 1

When X = 1, Y = 1, so P(Y = 1 | X = 1) = 1

When X = 2, Y = 1, so P(Y = 1 | X = 2) = 1

When X = 3, Y = 3, so P(Y = 3 | X = 3) = 1

The joint probability mass function P(X = x, Y = y) is given by the product of the individual probabilities:

P(X = x, Y = y) = P(X = x) \times P(Y = y | X = x)

The marginal probabilities for X and Y can be obtained by summing across the other variable:

P(X = x) =
`\sum_{\text{all } y} P(X = x, Y = y)`

P(Y = y) =
`\sum_{\text{all } x} P(X = x, Y = y)`

In this case, since the conditional probabilities for Y given X result in deterministic values, the joint PMF would reflect these direct associations. The marginal probabilities for X and Y can be directly obtained from the joint PMF.