Question

Let X be a continuous random variable with a PDF of the form

f_{X}(x) = c(1-x)\\ if x \in [0, 1] ,0 otherwise.

Find the following values

1 c

2. P(X = 1/2) -

3. P(X {1/k :k integer, k≥2}) =

4. P(X = 1/2)

Answer 1

To analyze the continuous random variable \( X \) with the given probability density function (PDF), we begin by finding the normalization constant \( c \).

### 1. Finding \( c \)

For the PDF to be valid, it must satisfy the property:

\[
\int_{-\infty}^{\infty} f_X(x) \, dx = 1
\]

Given that \( f_X(x) = c(1 - x) \) for \( x \in [0, 1] \) and \( 0 \) otherwise, we find:

\[
\int_0^1 c(1 - x) \, dx = 1
\]

Now we compute the integral:

\[
\int_0^1 (1 - x) \, dx = \left[ x - \frac{x^2}{2} \right]_0^1 = 1 - \frac{1}{2} = \frac{1}{2}
\]

Thus, we have:

\[
c \cdot \frac{1}{2} = 1 \quad \Rightarrow \quad c = 2
\]

### 2. \( P(X = 1/2) \)

For a continuous random variable, the probability of taking on any specific value is always zero:

\[
P(X = 1/2) = 0
\]

### 3. \( P(X \in \{1/k : k \text{ integer}, k \geq 2\}) \)

The set \( \{1/k : k \text{ integer}, k \geq 2\} \) contains countably infinite points (e.g., \( 1/2, 1/3, 1/4, \ldots \)). Since the probability of \( X \) taking any specific value is zero and this is a countably infinite set of points,

\[
P(X \in \{1/k : k \text{ integer}, k \geq 2\}) = 0
\]

### 4. \( P(X = 1/2) \)

As already calculated in step 2, for any specific value, including \( P(X = 1/2) \):

\[
P(X = 1/2) = 0
\]

### Summary of Results
1. \( c = 2 \)
2. \( P(X = 1/2) = 0 \)
3. \( P(X \in \{1/k : k \text{ integer}, k \geq 2\}) = 0 \)
4. \( P(X = 1/2) = 0 \)