Edexcel FS1 2022 June — Question 6

Exam BoardEdexcel
ModuleFS1 (Further Statistics 1)
Year2022
SessionJune
TopicProbability Generating Functions
TypeFind PGF from probability distribution
  1. The discrete random variable \(V\) has probability distribution
\(v\)234
\(\mathrm { P } ( V = v )\)\(\frac { 9 } { 25 }\)\(\frac { 12 } { 25 }\)\(\frac { 4 } { 25 }\)
  1. Show that the probability generating function of \(V\) is $$\mathrm { G } _ { V } ( t ) = t ^ { 2 } \left( \frac { 2 } { 5 } t + \frac { 3 } { 5 } \right) ^ { 2 }$$ The discrete random variable \(W\) has probability generating function $$\mathrm { G } _ { W } ( t ) = t \left( \frac { 2 } { 5 } t + \frac { 3 } { 5 } \right) ^ { 5 }$$
  2. Use calculus to find
    1. \(\mathrm { E } ( W )\)
    2. \(\operatorname { Var } ( W )\) Given that \(V\) and \(W\) are independent,
  3. find the probability generating function of \(X = V + W\) in its simplest form. The discrete random variable \(Y = 2 X + 3\)
  4. Find the probability generating function of \(Y\)
  5. Find \(\mathrm { P } ( Y = 15 )\)
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