An investment strategist models the daily value of a complex financial derivative, XXX, as
X=3V−4W X = 3V - 4W X=3V−4Wwhere V V\,V and W W\,W represent the daily returns of two independent indices. It is known that V∼N(45,62)V \sim N(45, 6^2)V∼N(45,62) and W∼N(20,32)W \sim N(20, 3^2)W∼N(20,32).
Determine P(X>75)P(X > 75)P(X>75).
The returns of five consecutive independent days V1,V2,V3,V4,V5 V_1, V_2, V_3, V_4, V_5\,V1,V2,V3,V4,V5 are each distributed as VVV. A composite variable S S\,S is defined as the sum of these returns, S=∑i=15ViS = \sum_{i=1}^{5} V_iS=∑i=15Vi.
A benchmark index T T\,T follows the distribution T∼N(240,σ2)T \sim N(240, \sigma^2)T∼N(240,σ2).
Given that P(S−T<−10)=0.6P(S - T < -10) = 0.6P(S−T<−10)=0.6 and that S S\,S and T T\,T are independent,
calculate the variance of TTT, giving your answer to one decimal place.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.