Question

A random variable \( X \) takes the values 0, 1, 2. Its mean is 1.2. If \( P(X = 0) = 0.3 \), then \( P(X = 1) = \)

• A. 0.1
• B. 0.5
• C. 0.2
• D. 0.4

Hint:

To find the probability of a value in a discrete random variable, use the formula for the mean and solve for the unknown probability.

Answer 1

The Correct Option is C

Step 1: Use the formula for the mean.
The mean \( \mu \) of a discrete random variable is given by: \[ \mu = \sum_{i} x_i P(X = x_i), \] where \( x_i \) are the possible values of the random variable and \( P(X = x_i) \) are the corresponding probabilities. We are given that the mean is 1.2, so: \[ 1.2 = 0 \times P(X = 0) + 1 \times P(X = 1) + 2 \times P(X = 2). \]
Step 2: Set up the equation.
We are also given that \( P(X = 0) = 0.3 \). Let \( P(X = 1) = p \) and \( P(X = 2) = 1 - 0.3 - p = 0.7 - p \). Substituting these into the equation for the mean, we get: \[ 1.2 = 1 \times p + 2 \times (0.7 - p). \] Simplifying: \[ 1.2 = p + 1.4 - 2p. \] Solving for \( p \), we get: \[ 1.2 - 1.4 = -p \quad \Rightarrow \quad p = 0.2. \]
Step 3: Conclusion.
Thus, \( P(X = 1) = 0.2 \), making option (C) the correct answer.