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I'm new to elliptic curve cryptography.
I just want to know in the case where I take a random number (private key) and find its associated public key, does the size of the public key depends upon the size of the random number chosen?. If so then how is it related? If not then what determines the size of the public key?
The size of the public key depends on the elliptic curve used. Any private key will produce a point on the curve, which is the same size – approximately 256 bits for 256-bit curves, for example.
Random numbers from a small range could be insecure, however. The secure way to generate the private key is to take it from the range $[1, l-1]$, where $l$ is the order of the base point. So in the end the size of the private key and the public key should be related, though not necessarily equal (if the cofactor is larger than 1).
