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

\[\Rightarrow \frac{k}{2} + \frac{k}{3} + k + \frac{k}{5} = 1\]
\[ \Rightarrow \frac{15k + 10k + 30k + 6k}{30} = 1\]
\[ \Rightarrow \frac{61k}{30} = 1\]
\[ \Rightarrow k = \frac{30}{61}\]

Now,

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