Moja sestra je umorna, a i ja sam umorna.

Because Serbian verbs change according to the subject.

• je = is (3rd person singular: he/she/it is)
• sam = am (1st person singular: I am)

So:

• Moja sestra je umorna = My sister is tired
• ja sam umorna = I am tired

Unlike English, Serbian does not use the same form for every subject.

Because it describes two different subjects:

• sestra = sister
• ja = I

In Serbian, adjectives usually agree with the noun or person they describe, so each clause needs its own adjective:

• Moja sestra je umorna
• ja sam umorna

You would not normally leave out the second umorna here.

Because umorna is the feminine singular form of the adjective tired.

Serbian adjectives agree in gender and number with the person or noun they describe.

• umoran = masculine singular
• umorna = feminine singular
• umorno = neuter singular
• umorni / umorne / umorna = plural forms, depending on gender/mixed group

Here:

• sestra is feminine, so umorna
• ja sam umorna shows that the speaker is also female

If a male speaker said this, he would say:

• Moja sestra je umorna, a i ja sam umoran.

Yes. It tells us the speaker is female.

The pronoun ja itself does not show gender, but the adjective does:

• ja sam umorna = said by a woman/girl
• ja sam umoran = said by a man/boy

So this sentence gives you extra information about who is speaking.

a i is a very natural way to mean something like:

• and also
• and ... too
• and I am too

In this sentence:

• Moja sestra je umorna, a i ja sam umorna.

the phrase a i connects the two ideas and adds the second one.

A rough natural English sense is:

• My sister is tired, and I am too.

Here, a links the clauses, and i adds the idea of also / too.

It can sometimes be just i ja..., but a i ja... is very common when contrasting or adding another person in a smooth, natural way.

Very roughly:

• i = and / also
• a = a linking word often used for and / while / but, depending on context
• a i together often gives the sense and also

So a i ja sam umorna sounds very natural after the first clause.

Yes, you could say Sestra je umorna, but it would mean Sister is tired or The sister is tired, depending on context.

Moja sestra specifically means my sister.

Serbian often omits possessives when the meaning is already obvious from context, but including moja makes it explicit.

So:

• Moja sestra je umorna = My sister is tired
• Sestra je umorna = Sister/My sister/The sister is tired depending on context

Yes, Serbian often drops subject pronouns because the verb already shows the person.

So instead of:

• ja sam umorna

you can often just say:

• sam umorna

However, in this sentence ja is useful because it adds emphasis and contrast:

• My sister is tired, and I am too.

The ja makes it clear that I am also tired.

It is in the nominative case, because it is the subject of the clause.

• sestra = nominative singular
• moja also appears in a form that agrees with sestra: feminine singular nominative

So:

• moja sestra = my sister as the subject

This is the basic dictionary-type form of the noun in this sentence.

The comma separates two clauses:

• Moja sestra je umorna
• a i ja sam umorna

Each part has its own subject and verb, so a comma is natural and standard here.

It works much like English punctuation in:

• My sister is tired, and I am too.

Yes, Serbian word order is more flexible than English, although some versions sound more natural than others.

The given sentence is very natural:

• Moja sestra je umorna, a i ja sam umorna.

You might also hear:

• Moja sestra je umorna, a ja sam i umorna.
  but this can shift emphasis and may sound less neutral in this exact context.

In beginner learning, it is best to keep the original order until you get used to how emphasis works.

The endings help show grammatical agreement.

In this sentence:

• moja ends in -a because it agrees with sestra (feminine singular)
• umorna ends in -a because it agrees with sestra
• the second umorna also ends in -a because the speaker is female

So these endings are not random. They tell you about:

• gender
• number
• sometimes case

Agreement is one of the most important features of Serbian grammar.

In Serbian, you would normally keep the adjective:

• Moja sestra je umorna, a i ja sam umorna.

English can say I am too and leave out tired, but Serbian usually repeats the adjective in this kind of sentence.

You may hear shorter conversational versions in speech, but for standard, clear Serbian, repeating umorna is normal and expected.