Question

A random variable X has the following probability functions: x: 0 1 2 3 4 5 6 7 8 p(x): lambda 3 lambda 5 lambda 7 lambda 9 lambda 11 lambda 13 lambda 15 lambda 17 lambda (i) Calculate the minimum value of lambda such that P(X<=3)>0*5.

Answer 1

$$P(X\leq3)=\lambda+3\lambda+5\lambda+7\lambda$$

$$=16\lambda>0.5$$

$$\lambda>1/8$$

There is no the exact minimum value of lambda such that $P(X\leq3)>0.5$

$$\lambda>1/8$$

Check

$$\lambda+3\lambda+5\lambda+7\lambda+9\lambda+11\lambda+13\lambda$$

$$+15\lambda+17\lambda=1$$

$$\lambda=1/81<1/8$$

Therefore the given problem has no solution.

The value of $\lambda$ must be $1/81$ to satisfy the probability function.