It is not until a character levels up that the dice would be rolled or the average would be taken.

Player's Handbook, page 12, top paragraph, right column:

"At 1st level, your character has 1 Hit Die, and the die type is determined by your class. You start with hit points equal to the highest roll of that die, as indicated in your class description. (You also add your Constitution modifier, which you'll determine in step 3.) This is also your hit point maximum."

A few paragraphs below, you see— under "Building Bruenor, Step 2":

"As a 1st-level fighter, Bruenor has 1 Hit Die— a d10 —and starts with hit points equal to 10 + his Constitution modifier." and on page 13, we see his Con modifier is +3.

So you don't actually roll or "take" anything, at first level. You get the maximum available for the class.

Player's Handbook, page 15, left column, under "Beyond 1st Level":

"Each time you gain a level, you gain 1 additional Hit Die. Roll that Hit Die, add your Constitution modifier to the roll, and add the total to your hit point maximum. Alternatively, you can use the fixed value shown in your class entry, which is the average resuly of the die roll (rounded up)."