Question

Let X be a random variable which assumes values x₁​, x₂​, x₃​, x₄​ such that  2P(X = x_1​) = 3P(X = x₂​) = P(X = x₃​) = 5P(X = x₄​).

Find the probability distribution of X.

Answer 1

Given 2P(X = x₁​) = 3P(X = x₂​) = P(X = x_3​) = 5P(X = x_4​) = k

\(\displaystyle\sum^4_{i=1} p(X = x_i) = 1\)

⇒ P(X = x₁​) + P(X = x₂​) + P(X = x_3​) + P(X = x₄​) =1

⇒ \(\frac k2 + \frac k3 + k+\frac k5 = 1\)

⇒ \(\frac{15k + 10 k+ 30k + 6k}{30} = 1\)

⇒ \(k = \frac{30}{61}\)

P(X = x₁​) = \(\frac{15}{61}\)

P(X = x₂​) = \(\frac{10}{61}\)

P(X = x₃​) = \(\frac{30}{61}\)

P(X = x₄​) = \(\frac{6}{61}\)