Question

1)Find the constant c so that a random variable X is defined by taking on values 1 , 2 , … , n , … with probabilities c*(1/4)^n.
2) Find the constant c so that a random variable X is defined assigning any interval [a,b] probability c/π⋅∫ba (1/x^2+1)) dx

Answer 1

for part one i guess is
1) 1-(1/4)= 3/4
(1/4)(3/4) = 1/3
c*1/3=1 =>c=3

Answer 2

for part one I don't know for sure how I'm support to do, my guess is
let 3t=x
3dt=dx

Answer 3

\[\int\limits_{-\infty}^{\infty} \frac{ 1 }{ x^2+ 1}\]

Answer 4

\[\int\limits_{-\infty}^{\infty} \frac{ 1 }{ 9t^2+ 1} * 3 dt\]