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The oxidation number of ##H## is +1. The oxidation number of ##O## is -1.
You assign to the in a compound by using the Rules for Oxidation Numbers.
The important rules for this problem are:
The oxidation number of ##H## is +1 but it is -1 in when combined with less electronegative elements.
The oxidation number of ##O## in is usually -2 but it is -1 in peroxides.
The sum of the oxidation numbers of all the atoms in a neutral compound is 0.
The oxidation number of ##H## is +1 (Rule 1).
If you know that ##H_2O_2## is hydrogen peroxide you can immediately assign oxygen the ON = -1.
If you don’t know that ##H_2O_2## is hydrogen peroxide you can use Rule 3.
The oxidation number of ##H## is +1. For two ##H## atoms the sum is +2.
Since ##H_2O_2## has no charge the total oxidation number of the two ##O## atoms must be -2.
The oxidation number of one oxygen atom must be -1.