Question

A random variable X takes the values 0, 1, 2, 3 and its mean is 1.3. If P (X = 3) = 2 P (X = 1) and P (X = 2) = 0.3, then P (X = 0) is
(a) 0.1
(b) 0.2
(c) 0.3
(d) 0.4

Answer 1

(d) 0.4

Let:
P(X = 0) = m
P(X = 1) = k.

Now,
P(X = 3) = 2k

xi	pi	pixi
0	m	0
1	k	k
2	0.3	0.6
3	2k	6k

Mean = ${\sum }_{}^{}$p_ix_i

$$\begin{gathered} 0+k+0.6+6k=1.3 \\ \Rightarrow 7k=1.3-0.6 \\ \Rightarrow k=\frac{0.7}{7}=0.1 \end{gathered}$$

We know that the sum of probabilities in a probability distribution is always 1.

∴ P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3) = 1

$$\begin{gathered} \Rightarrow m+0.1+0.3+0.2=1 \\ \Rightarrow m+0.6=1 \\ \Rightarrow m=0.4 \end{gathered}$$