Question

A random variable X takes values 0, 1, 2, 3 and its mean is 1.3. If $P(X=3)=2P(X=1)$ and $P(X=2)=0.3$, then find $P(X=0)$.

• A. \(\frac{1}{5} \)
• B. \(\frac{2}{5} \)
• C. \(\frac{3}{5} \)
• D. \(\frac{4}{5} \)

Hint:

Always start by checking if the sum of probabilities is 1; this often provides the first necessary constraint to solve for unknown variables.

Answer 1

The Correct Option is B

Step 1: Introduce unknown probabilities.
Let $P(X=1)=p$. Then $P(X=3)=2p$ and $P(X=2)=0.3$ are given. Let $P(X=0)=p_0$.
Step 2: Use total probability equals one.
\[ p_0+p+0.3+2p=1\ \Longrightarrow\ p_0+3p=0.7. \]
Step 3: Write the mean equation.
The mean is $\sum xP(x)=0\cdot p_0+1\cdot p+2\cdot0.3+3\cdot2p=1.3$.
Step 4: Simplify the mean equation.
This becomes $p+0.6+6p=1.3$, so $7p=0.7$, giving $p=0.1$.
Step 5: Back-substitute for $p_0$.
From Step 2, $p_0+3(0.1)=0.7$, so $p_0=0.7-0.3=0.4$.
Step 6: Express as a fraction.
$p_0=0.4=\tfrac{4}{10}=\tfrac25$, which is option 2.
\[ \boxed{P(X=0)=\tfrac25} \]