Should the battery of the iPad1 drain while it is off? It appears to be happening to me.

When you say 'off' do you mean this - or is the iPad in Sleep mode.

When you briefly touch the Power key you put the iPad into Sleep mode and, then, several things are still running - such as WiFi and 3G, if they're turned on in 'Settings' - and your iPad may occasionally check for email - again if you've asked for this in Settings.

If you've really switched the iPad off - by holding down the Power key for a couple of seconds and then sliding the control to authorise the iPad to switch off, then the battery drain should be very small.

You can read Apple's official line on the iPad battery here

Apple - Batteries - iPad

Tim

I guess I am lucky, at night I put it in sleep mode and in the morning it still reads the same as the night before.