How to redistribute weightage proportionally?

Question

I want to increase proportional increase weightage.

For example, I have weights

w1 = 0.4
w2 = 0.3
w3 = 0.2
w4 = 0.1

with a constraint that the total sum of the weights should always be 1 i.e w1 + w2 + w3 + w4 = 1

Now, w4 has been removed. I want to proportionally increase w1, w2 and w3. so that

w1 -> 0.44
w2 -> 0.33
w3 -> 0.22

Also, the total sum of the weights should always be 1 i.e w1 + w2 + w3 = 1.

How to do it mathematically?

Answer 1

Let $v_i$ be the updated $w_i$,

then $$v_i = \frac{w_i}{\sum_{j\neq 4}w_j}=\frac{w_i}{1-w_4}$$

In general, if $w_j$ is removed, then

$$v_i =\frac{w_i}{1-w_j}$$

Notice that if $w_j=1$, then all the remaining $w_i=0$, $i \neq j$.