Stochastics: discrete random variables

[sunrah]

Homework Statement

X₁ and X₂ are two independent discrete random variables with
P(X₁ = k) = c3^-k
P(X₂ = k) = d4^-k

for k in natural numbers and where X₁, X₂ in natural numbers is almost always valid. 0 is not include in N.

Find constants c and d.

Homework Equations

The Attempt at a Solution

Since I'm joining this class late in the semester I don't know where to begin. Any help is appreciated!

 

[Curious3141]

Hint: what must the total probability of all possible events (in the entire sample space) be?

Another hint: geometric series.

 

[sunrah]

Hint: what must the total probability of all possible events (in the entire sample space) be?

Another hint: geometric series.

do you mean sum to infinity?

\Sigma^{\infty}_{K=1} c3^{-k} = 1

I see that the series (s_n) = c3^-k converges.

 

[Curious3141]

do you mean sum to infinity?

\Sigma^{\infty}_{K=1} c3^{-k} = 1

I see that the series (s_n) = c3^-k converges.

So what's the sum?

 

[sunrah]

So what's the sum?

ya, so in this case (s_n) = 3^-k converges against 1/2. So c = 2

thanks!

 

[Curious3141]

ya, so in this case (s_n) = 3^-k converges against 1/2. So c = 2

thanks!

I would've simply said (c)(\frac{1}{2}) = 1 \Rightarrow c = 2. But, yes, you've got the idea.