Question

The distribution function \( F(X) \) of a discrete random variable \( X \) is given. Then \( P[X = 4] + P[X = 5] \):
\[ \begin{array}{|c|c|c|c|c|c|c|} \hline X & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline F(X = x) & 0.2 & 0.37 & 0.48 & 0.62 & 0.85 & 1 \\ \hline \end{array} \]

• A. \(0.14\)
• B. \(0.85\)
• C. \(0.37\)
• D. \(0.23\)

Hint:

Ensure all probabilities add up to 1 for valid distributions.

Answer 1

The Correct Option is C

Step 1: Understand the distribution function \( F(X) \):
The provided \( F(X) \) is the cumulative distribution function (CDF), indicating the probability that the random variable \( X \) is less than or equal to a given value \( x \). 

Step 2: Calculate \( P[X = 4] \): 
The probability \( P[X = 4] \) is calculated as the difference between \( F(4) \) and \( F(3) \). 
From the table, \( F(4) = 0.62 \) and \( F(3) = 0.48 \). 
Therefore, \( P[X = 4] = 0.62 - 0.48 = 0.14 \). 

 Step 3: Calculate \( P[X = 5] \): 
Similarly, \( P[X = 5] \) is calculated as \( F(5) - F(4) \). 
From the table, \( F(5) = 0.85 \) and \( F(4) = 0.62 \). 
Therefore, \( P[X = 5] = 0.85 - 0.62 = 0.23 \). 

Step 4: Sum the probabilities: 
The total probability is the sum of \( P[X = 4] \) and \( P[X = 5] \). 
\( P[X = 4] + P[X = 5] = 0.14 + 0.23 = 0.37 \). 

Final Answer: \[ \boxed{0.37}. \]