# How to find the distribution of the weighted sum of independent Bernoulli random variables for positive non-integer weights

How do I find the distribution of the weighted sum of independent Bernoulli random variables if the weights are non-negative real numbers?

I have N number of independent Bernoulli distributed random variables lets say $X_1, X_2, X_3...X_N$ and suppose I have a set of weights $W_1, W_2, W_3,...,W_N$ which are non-negative real numbers and the sum of all the weights is equal to $N$.

Then I need to find the distribution of the random variable $Z$ which is the weighted sum of the $N$ independent Bernoulli random variables i.e, $Z =\sum_i^N W_i*X_i$, where $i=1,2,\cdots,N$.

• This is conditional on the $W$'s? Do the $X_i$ all share the same $p$? – Glen_b Aug 18 '14 at 2:01
• No, all Xi have different probability P and I am assuming the weight parameter W is constant for all Xis it doesnt change. If Xi is highly probable then corresponding weight Wi will take large value between 0 and N. Thanks inadvance, Glen_b – nuria Aug 20 '14 at 6:52
• Should answers be exact or approximate? – Glen_b Aug 20 '14 at 8:04
• Okay, but I don't think you'll much like the answer. – Glen_b Aug 21 '14 at 2:00
• When you say the weights are dependent, you're indicating that the weights are random variables rather than constants, unlike the situation here? – Glen_b Oct 25 '18 at 5:58