Question

The random variable J has a geometric distribution and it is given that, 𝑃(𝐽=5)/𝑃(𝐽=7) = 25/16 .

Find 𝑃(𝐽 = 4)

Answer 1

For the geometric distribution

$$P(J=j) = p(1-p)^{j-1}$$

Therefore

$$\frac{P(J=5)}{P(J=7)} = \frac{p(1-p)^{5-1}}{p(1-p)^{7-1}}=\frac{25}{16}$$

$$\frac{(1-p)^{4}}{(1-p)^{6}}=\frac{25}{16}$$ $$(1-p)^{-2}=\frac{25}{16}$$ $$-2 * log (1-p)=log (\frac{25}{16})$$ $$-2 * log (1-p)=0.1938$$ $$log (1-p)=-0.0969$$ $$1-p = e^{-0.0969}$$ $$p=1-0.90763767804293$$ $$p=0.0924$$

The PDF is therefore

$$P(J=j) = 0.0924(0.9076)^{j-1}$$ $$P(J=4) = 0.0924(0.9076)^{4-1}$$ $$P(J=4) =0.0691$$