A random variable X takes the 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 \(P(X=0)\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0.4
Let \(P(X=1) = p\). Then \(P(X=3) = 2p\) and \(P(X=2) = 0.3\).
The mean is \(\sum x_iP(x_i)\):
\(0 \cdot P(X=0) + 1(p) + 2(0.3) + 3(2p) = 1.3\)
\(7p + 0.6 = 1.3\)
\(p = 0.1\)
So \(P(X=1) = 0.1\) and \(P(X=3) = 0.2\). The probabilities add up to 1:
\(P(X=0) = 1-(0.1+0.3+0.2)\)
\(P(X=0) = 0.4\)