For the following distribution function `F (x) `of a r.v X
`{:(x,1,2,3,4,5,6),(F(x),0.2,0.37,0.48,0.62,0.85,1):}`
`p(3 lt x le 5)=`

Options

A. `0.48`
B. `0.37`
C. `0.27`
D. `1.47`

Answer

To find the probability \( P(3 < X \leq 5) \) from the given distribution function \( F(x) \), we need to follow these steps:

### Step 1: Understand the given distribution function
The distribution function \( F(x) \) is given as:
- \( F(1) = 0.2 \)
- \( F(2) = 0.37 \)
- \( F(3) = 0.48 \)
- \( F(4) = 0.62 \)
- \( F(5) = 0.85 \)
- \( F(6) = 1 \)

### Step 2: Calculate the probabilities for each value of \( X \)
To find the probability \( P(X = x) \) for each value of \( X \), we can use the differences between the cumulative probabilities:
- \( P(X = 1) = F(1) = 0.2 \)
- \( P(X = 2) = F(2) - F(1) = 0.37 - 0.2 = 0.17 \)
- \( P(X = 3) = F(3) - F(2) = 0.48 - 0.37 = 0.11 \)
- \( P(X = 4) = F(4) - F(3) = 0.62 - 0.48 = 0.14 \)
- \( P(X = 5) = F(5) - F(4) = 0.85 - 0.62 = 0.23 \)
- \( P(X = 6) = F(6) - F(5) = 1 - 0.85 = 0.15 \)

### Step 3: Identify the relevant probabilities for \( P(3 < X \leq 5) \)
The probability \( P(3 < X \leq 5) \) includes the values of \( X = 4 \) and \( X = 5 \):
- \( P(X = 4) = 0.14 \)
- \( P(X = 5) = 0.23 \)

### Step 4: Calculate the total probability
Now, we add the probabilities of \( X = 4 \) and \( X = 5 \):
\[ P(3 < X \leq 5) = P(X = 4) + P(X = 5) = 0.14 + 0.23 = 0.37 \]

### Final Answer
Thus, the probability \( P(3 < X \leq 5) \) is \( \boxed{0.37} \).

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