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

A random variable \(X\) has a proper probability density function \(f(x)\) that satisfies
\[\begin{cases}\displaystyle\sum_{x\in \mathbb{S}}f(x)=1&\text{if }X\text{ is a discrete RV}\\[1ex]
\displaystyle\int_\mathbb{S}f(x)\,dx=1&\text{if }X\text{ is a continuous RV}\end{cases}\]where \(\mathbb{S}\) denotes the sample space or support, whatever you like to call it.

For (1), you have to find \(c\) such that
\[\sum_{n=1}^\infty c\left(\frac{1}{4}\right)^n=1\]which is easy enough if you recall the known sum,
\[\sum_{n=1}^\infty r^{n-1}=\frac{1}{1-r}\]
For (2), you need to find \(c\) such that the given integral evaluates to \(1\), i.e.
\[\frac{1}{\pi}\int_a^b\frac{dx}{x^2+1}=\frac{1}{c}\]

Answer 2

i see

Answer 3

let 3t=x
3dt=dx