The probability distribution of random variable X with two missing probabilities `p_(1)` and `p_(2)` is given below
`{:(X, P(X)),(1, k), (2, p_(1)),(3, 4k),(4, p_(2)),(5, 2k):}`
It is further given that `P(X le 2) = 0.25 and P(X ge 4) = 0.35`. Consider the following statements
1. `p_(1) = p_(2)`
2. `p_(1) + p_(2) = P(X = 3)`
which of the statements given above is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the problem, we need to analyze the given probability distribution and the conditions provided. Let's break it down step by step.

### Step 1: Understand the Probability Distribution
We have the following probability distribution for the random variable \( X \):
- \( P(X = 1) = k \)
- \( P(X = 2) = p_1 \)
- \( P(X = 3) = 4k \)
- \( P(X = 4) = p_2 \)
- \( P(X = 5) = 2k \)

### Step 2: Use the Total Probability Rule
The sum of all probabilities must equal 1:
\[ k + p_1 + 4k + p_2 + 2k = 1 \]
This simplifies to:
\[ 7k + p_1 + p_2 = 1 \quad \text{(Equation 1)} \]

### Step 3: Analyze the Given Conditions
We are given two conditions:
1. \( P(X \leq 2) = 0.25 \)
2. \( P(X \geq 4) = 0.35 \)

#### Condition 1: \( P(X \leq 2) = 0.25 \)
This means:
\[ P(X = 1) + P(X = 2) = k + p_1 = 0.25 \quad \text{(Equation 2)} \]

#### Condition 2: \( P(X \geq 4) = 0.35 \)
This means:
\[ P(X = 4) + P(X = 5) = p_2 + 2k = 0.35 \quad \text{(Equation 3)} \]

### Step 4: Solve the Equations
Now we have three equations:
1. \( 7k + p_1 + p_2 = 1 \) (Equation 1)
2. \( k + p_1 = 0.25 \) (Equation 2)
3. \( p_2 + 2k = 0.35 \) (Equation 3)

From Equation 2, we can express \( p_1 \) in terms of \( k \):
\[ p_1 = 0.25 - k \quad \text{(Equation 4)} \]

From Equation 3, we can express \( p_2 \) in terms of \( k \):
\[ p_2 = 0.35 - 2k \quad \text{(Equation 5)} \]

### Step 5: Substitute into Equation 1
Now substitute Equations 4 and 5 into Equation 1:
\[ 7k + (0.25 - k) + (0.35 - 2k) = 1 \]
Simplifying this:
\[ 7k - k - 2k + 0.25 + 0.35 = 1 \]
\[ 4k + 0.60 = 1 \]
\[ 4k = 1 - 0.60 \]
\[ 4k = 0.40 \]
\[ k = 0.10 \]

### Step 6: Find \( p_1 \) and \( p_2 \)
Now we can find \( p_1 \) and \( p_2 \):
Using Equation 4:
\[ p_1 = 0.25 - k = 0.25 - 0.10 = 0.15 \]

Using Equation 5:
\[ p_2 = 0.35 - 2k = 0.35 - 2(0.10) = 0.35 - 0.20 = 0.15 \]

### Step 7: Verify the Statements
1. **Statement 1**: \( p_1 = p_2 \)
- True, since \( p_1 = 0.15 \) and \( p_2 = 0.15 \).

2. **Statement 2**: \( p_1 + p_2 = P(X = 3) \)
- \( p_1 + p_2 = 0.15 + 0.15 = 0.30 \)
- \( P(X = 3) = 4k = 4(0.10) = 0.40 \)
- This statement is false.

### Conclusion
Only Statement 1 is correct.