Calculate E(X) from cumulative distribution

The cumulative distribution function F(X) is given, requiring conversion to probability distribution P(X=x) before calculating E(X).

2 questions

Edexcel S1 2022 October Q4
  1. The cumulative distribution function of the discrete random variable \(W\), which takes only the values 6,7 and 8 , is given by
$$F ( W ) = \frac { ( w + 3 ) ( w - 1 ) } { 77 } \text { for } w = 6,7,8$$ Find \(\mathrm { E } ( W )\)
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Edexcel S1 Q5
5. The discrete random variable \(Y\) has the following cumulative distribution function.
\(y\)01234
\(\mathrm {~F} ( Y )\)0.050.150.350.751
  1. Write down the probability distribution of \(Y\).
  2. Find \(\mathrm { P } ( 1 \leq Y < 3 )\).
  3. Show that \(\mathrm { E } ( Y ) = 2.7\)
  4. Find \(\mathrm { E } ( 2 Y + 4 )\).
  5. Find \(\operatorname { Var } ( Y )\).