The probability distribution of a random discrete variable is given

If it is known that P(X=1) is the mean of P(X=0) and P(X=2).

Then the value of $r$ is:

0

The correct answer is Option (1) → 0 **

Given probabilities:

$P(X=-1)=0.1,\; P(X=0)=p,\; P(X=1)=0.3,\; P(X=2)=q,\; P(X=3)=r$

The given condition:

$P(X=1)$ is the mean of $P(X=0)$ and $P(X=2)$

This means:

$0.3=\frac{p+q}{2}$

So:

$p+q=0.6$

Using total probability = 1:

$0.1 + p + 0.3 + q + r = 1$

$p + q + r = 0.6$

Since $p+q=0.6$:

$0.6 + r = 0.6$

$r = 0$

Final values:

$p + q = 0.6$ (infinitely many pairs possible)

$r = 0$