Question

Let X be a random variable which assumes values x₁, x₂, x₃, x₄ such that 2P (X = x₁) = 3P (X = x₂) = P (X = x₃) = 5 P (X = x₄). Find the probability distribution of X.

Answer 1

Let P (X = x₃) = k. Then,
P (X = x₁) = $\frac{k}{2}$

P (X = x₂) = $\frac{k}{3}$

P (X = x₄) = $\frac{k}{5}$

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

∴ P (X = x₁) + P (X = x₂) + P (X = x₃) + P (X = x₄) = 1

$$\begin{gathered} \Rightarrow \frac{k}{2}+\frac{k}{3}+k+\frac{k}{5}=1 \\ \Rightarrow \frac{15k+10k+30k+6k}{30}=1 \\ \Rightarrow \frac{61k}{30}=1 \\ \Rightarrow k=\frac{30}{61} \end{gathered}$$

Now,

xi	pi
x1	$\frac{k}{2}$ = $\frac{15}{61}$
x2	$\frac{k}{3}$ = $\frac{10}{61}$
x3	k = $\frac{30}{61}$
x4	$\frac{k}{5}$= $\frac{6}{61}$