We consider the problem of providing full coverage of a planar region with sensors. Likewise, we consider the connectivity of the sensor network of directional antennas formed by sensors in this region. Suppose that n sensors with coverage angle (also known as beam width) a(n), and reachability radius r(n) are thrown randomly and independently with the uniform distribution in the interior of the unit square. Let pin) be the probability that a given sensor is active. We prove that if p(n) = Ω ((log)(n/r(n)~2 sin~2 (α)(n)/4))/(nr(n)~2 α (n) sin (α(n)/4)) then the Probability the sensors pro-vide full coverage of the unit square is at least 1 - n~(-O(1)). Likewise, we consider the connectivity of the resulting sensor network. We show that if p(n) = Ω((log(n/r(n)~2 sin~2 (α(n)/4))/(nr(n)~2 α(n) sin~2 (α(n)/4)) then the probability that a connected subnetwork of sensors provides full coverage is at least 1 - n~~(-O(1)).
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