Proof and derivation of E(X) and Var(X)

Questions requiring formal proof from first principles of the binomial mean and variance formulas, or deriving variance using E(X(X-1)).

2 questions

AQA S3 2012 June Q7
7
  1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
    1. Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
    2. Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find, in terms of \(n\) and \(p\), an expression for \(\operatorname { Var } ( X )\).
  2. The mode, \(m\), of \(X\) is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
    1. Use the first inequality to show that $$m \leqslant ( n + 1 ) p$$
    2. Given that the second inequality results in $$m \geqslant ( n + 1 ) p - 1$$ deduce that the distribution \(\mathrm { B } ( 10,0.65 )\) has one mode, and find the two values for the mode of the distribution \(B ( 35,0.5 )\).
  3. The random variable \(Y\) has a binomial distribution with parameters 4000 and 0.00095 . Use a distributional approximation to estimate \(\mathrm { P } ( Y \leqslant k )\), where \(k\) denotes the mode of \(Y\).
    (3 marks)
OCR MEI Further Statistics Major 2023 June Q11
11 The random variable \(X\) takes the value 1 with probability \(p\) and the value 0 with probability \(1 - p\).
  1. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    • The random variable \(Y \sim \mathrm {~B} ( 50,0.2 )\) has mean \(\mu\) and variance \(\sigma ^ { 2 }\).
    Use the results of part (a) to prove that
    • \(\mu = 10\)
    • \(\sigma ^ { 2 } = 8\).