Answer:
The normalization property of the distribution law states that the sum of probabilities of all possible outcomes must equal 1. In this case, we have:
P(X=k) = c/2^k, for k = 0, 1, 2, ...
To find the value of the constant c, we need to use the normalization property:
∑ P(X=k) = ∑ c/2^k = c/2^0 + c/2^1 + c/2^2 + ... = 1
To simplify this expression, we can use the formula for the sum of an infinite geometric series:
∑ a*r^n = a/(1-r), where a is the first term, r is the common ratio, and n goes from 0 to infinity.
In this case, a = c, r = 1/2, and n goes from 0 to infinity. So we have:
∑ P(X=k) = ∑ c/2^k = c/2^0 + c/2^1 + c/2^2 + ... = c/(1-1/2) = 2c
Setting this expression equal to 1, we get:
2c = 1
c = 1/2
Therefore, the value of the constant c is 1/2.
Explanation: