Question

Need help with geometric distribution

Answer 1

The image is kinda blurry, but from what I can gather, the notation in the attachment means

$p_X[x_k]\equiv\mathbb P(X=k)=\begin{cases}(1-p)^{k-1}p&\text{for }k=1,2,\ldots\\0&\text{otherwise}\end{cases}$

Then

a) $p_X[x_k\le4]\equiv\mathbb P(X\le4)=\mathbb P(X=1)+\mathbb P(X=2)+\mathbb P(X=3)+\mathbb P(X=4)$

b) $p_X[1$

c) $p_X[1\le x_k\le2]\equiv\mathbb P(1\le X\le2)=\mathbb P(X=1)+\mathbb P(X=2)$