How do I activate my iPhone when nobody knows the old password?

Have you tried apples iForgot service? see this article and scroll down a bit

If you forgot your Apple ID password – Apple Support (AU)

This article can also be of help:

How to remove Activation Lock - Apple Support

Ask your sister to -> Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac

Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac - Apple Support

Or check her Apple ID via her email box, see this Apple article: If you forgot your Apple ID

If you forgot your Apple ID email address or phone number - Apple Support                  

OR see -> Having trouble signing in?

https://iforgot.apple.com/password/verify/appleid

If you sister has forgotten her password? -> See this Apple article: If you forgot your Apple ID password

If you forgot your Apple ID password - Apple Support

Then see this Apple article: What to do after you change your Apple ID or password

What to do after you change your Apple ID or password - Apple Support

Each Apple device is linked/associated with an Apple ID of the person who did the setup.

Activation Lock prevents anyone who is not the owner from using this iPhone.

To unlock the Phone, you need the Apple ID and password of the previous owner (your sister), that were used during set up.

So you have to contact the previous owner.

The previous owner (your sister) has to read -> this Apple article: What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch

What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support

Then the previous owner has to remove the iPhone from his account, see this Apple article:

Check your Apple ID device list to see where you're signed in

Check your Apple ID device list to see where you're signed in - Apple Support

Then you can sign in with own your Apple ID.

See the following Apple articles:

Activation Lock for iPhone, iPad, and iPod touch

Activation Lock for iPhone, iPad, and iPod touch - Apple Support

See -> How to remove Activation Lock

How to remove Activation Lock – Apple Support (UK)

You can contact Apple support for help, -> Get Support (apple.com)

You have to show proof of ownership of your product, provide all documentation showing your purchase of the device.

If you do not have additional proof of ownership, contact the reseller. They may be able to provide the original purchase document.

Once Apple receive your proof of ownership, Apple will further review your request and provide an update.

Hi there!

You can ask your sister to recover their account/reset Apple ID password by following the steps in this article

If you have forgotten your Apple ID password – Apple Support (UK)

Depending on how their account was set up, they might have the option to request for account recovery if all methods to reset the password has been exhausted

How to use account recovery when you can’t reset your Apple ID password - Apple Support

If they can no longer recover their account, they can rquest the activation lock be removed by submitting a proof of purchase documentation at this link:  https://al-support.apple.com/#/kbase