Suppose that the r.v X has possible values 1,2,…. and 𝑃(𝑋 = 𝑥𝑗) = 1 2 𝑗 ,𝑗 = 1,2, … … … .. 
 a. compute𝑃( 𝑥 𝑖𝑠 𝑜𝑑𝑑)             b. compute 𝑃(𝑥 ≥ 4)        c. compute 𝑃(𝑥 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 2)


Given the random variable \( X \) with possible values \( 1, 2, 3, \dots \), and the probability mass function \( P(X = x_j) = \frac{1}{2^j} \), we can now compute the following probabilities:

### a. Compute \( P(X \text{ is odd}) \)
Here, we need to find the probability that \( X \) takes on an odd value. The odd values of \( X \) are \( 1, 3, 5, \dots \).

The probability that \( X \) is odd is given by the sum of the probabilities for all odd values of \( X \):

\[
P(X \text{ is odd}) = P(X = 1) + P(X = 3) + P(X = 5) + \dots 
\]

Since \( P(X = j) = \frac{1}{2^j} \), we can express this as an infinite series:

\[
P(X \text{ is odd}) = \frac{1}{2^1} + \frac{1}{2^3} + \frac{1}{2^5} + \dots
\]

This is a geometric series with the first term \( a = \frac{1}{2} \) and a common ratio \( r = \frac{1}{4} \). The sum of an infinite geometric series is given by:

\[
S = \frac{a}{1 - r}
\]

Substituting \( a = \frac{1}{2} \) and \( r = \frac{1}{4} \):

\[
P(X \text{ is odd}) = \frac{\frac{1}{2}}{1 - \frac{1}{4}} = \frac{\frac{1}{2}}{\frac{3}{4}} = \frac{1}{2} \times \frac{4}{3} = \frac{2}{3}
\]

Thus, the probability that \( X \) is odd is:

\[
P(X \text{ is odd}) = \frac{2}{3}
\]

---

### b. Compute \( P(X \geq 4) \)
We need to compute the probability that \( X \) takes a value greater than or equal to 4. This is given by the sum of probabilities for all \( X \geq 4 \):

\[
P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6) + \dots
\]

This can be written as:

\[
P(X \geq 4) = \frac{1}{2^4} + \frac{1}{2^5} + \frac{1}{2^6} + \dots
\]

Again, this is a geometric series with the first term \( a = \frac{1}{16} \) and a common ratio \( r = \frac{1}{2} \). The sum of the series is:

\[
S = \frac{a}{1 - r}
\]

Substituting \( a = \frac{1}{16} \) and \( r = \frac{1}{2} \):

\[
P(X \geq 4) = \frac{\frac{1}{16}}{1 - \frac{1}{2}} = \frac{\frac{1}{16}}{\frac{1}{2}} = \frac{1}{16} \times 2 = \frac{1}{8}
\]

Thus, the probability that \( X \geq 4 \) is:

\[
P(X \geq 4) = \frac{1}{8}
\]

---

### c. Compute \( P(X \text{ is divisible by 2}) \)
We need to compute the probability that \( X \) is divisible by 2. The values of \( X \) that are divisible by 2 are \( 2, 4, 6, 8, \dots \).

The probability that \( X \) is divisible by 2 is the sum of the probabilities for all even values of \( X \):

\[
P(X \text{ is divisible by 2}) = P(X = 2) + P(X = 4) + P(X = 6) + \dots
\]

This can be written as:

\[
P(X \text{ is divisible by 2}) = \frac{1}{2^2} + \frac{1}{2^4} + \frac{1}{2^6} + \dots
\]

This is another geometric series with the first term \( a = \frac{1}{4} \) and common ratio \( r = \frac{1}{4} \). The sum of the series is:

\[
S = \frac{a}{1 - r}
\]

Substituting \( a = \frac{1}{4} \) and \( r = \frac{1}{4} \):

\[
P(X \text{ is divisible by 2}) = \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{4} \times \frac{4}{3} = \frac{1}{3}
\]

Thus, the probability that \( X \) is divisible by 2 is:

\[
P(X \text{ is divisible by 2}) = \frac{1}{3}
\]

---

### Summary of Results:
- \( P(X \text{ is odd}) = \frac{2}{3} \)
- \( P(X \geq 4) = \frac{1}{8} \)
- \( P(X \text{ is divisible by 2}) = \frac{1}{3} \)

---

Do you want more details or have any questions?

Here are 5 related questions to explore:
1. How do you compute probabilities for a random variable following other probability distributions, like Poisson?
2. What is the geometric series formula and why does it apply here?
3. How would you calculate the expectation \( E(X) \) for this random variable?
4. How can you compute \( P(X \leq 3) \) given the same distribution?
5. What is the cumulative distribution function (CDF) for this type of discrete random variable?

**Tip:** The geometric series is often used in probability to compute sums involving exponentially decreasing probabilities.

Learn how to compute probabilities for a random variable with a geometric probability mass function. Solve for P(X is odd), P(X ≥ 4), and P(X is divisible by 2) using geometric series and probability theory principles. Suitable for advanced high school students or undergraduate level.

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