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Further Stats Year 2
A-Level Further Maths- Edexcel
| Question | Answer |
|---|---|
| When using successive, independent trials with the same probability, you can use what distribution to model the number of trials before a single success? | Geometric |
| What is the P(X=x) | X~Geo(p) | p(1-p) ^ (x-1) |
| What is P(X≤x) | X~Geo(p) | 1-(1-p)^x |
| What is P(X≥x) | X~Geo(p) | (1-p)^(x-1) |
| If X~Geo(p) then what is E(X)? | 1/p |
| If X~Geo(p) then what is Var(X)? | 1-p / p^2 |
| In X~NB(r,p) what are r and p? | r: no. of successes needed p: probability of success |
| What is P(X=x) | X~NB(r,p) | ((x-1)C(r-1)) p^r (1-p)^(x-r) |
| If X~NB(r,p) then what is E(X)? | r/p |
| If X~NB(r,p) then what is Var(X)? | r(1-p) / p^2 |
| Given a random sample of size n taken from any distribution with mean μ and variance σ^2, the sample mean is approximately distributed by... | N(μ, σ^2/n) |
| What should you use when trying to find a minimum/ maximum sample size? | the standard normal distribution |
| For chi squared tests using geometric distribution, how do you work out the expected frequency of the final column? | Total - all other frequencies |
| What is Gx(t) | ΣP(X=x)t^x (ie P(X=0) + t P(X=1) + t^2 P(X=2).....) |
| What is Gx(1) = | 1 |
| How do we derive the PGF of a binomial distribution? | Write the function in the form of a binomial expansion and then factorise |
| How do we derive the PGF of a poisson distribution? | Write the function, take out a factor of e^something, rewrite whats left as e^something using taylor series |
| How do we derive the PGF of a geometric distribution? | Write the function, apply the formula for the sum to infinity of a geometric progression |
| How do we derive the PGF of a negative binomial distribution? | In the sum make t^x -> t^x-r t^r. Combine the terms to the same power. Take out anything that does not involve x. Use the general result given. Substitute in 1-p. |
| What is E(X) of a PGF? | G'x (1) |
| What is Var(X) of a PGF? | G''x(1) + G'x(1) - (G'x(1))^2 |
| What is the PGF of Z if Z = X+Y | Gx(t) x Gy(t) |
| What is the PGF of Y if Y= aX+b | t^b Gx(t^a) |
| Using a PGF what is P(X=x) | 1/x! G(differentiated x times)x(0) |
| What is a type I error? | Reject H0 when H0 is true. Probability = actual significance level |
| What is a type II error? | Accept H0 when H0 is false. |
| What is the probability of a type I error for a continuous distribution? | Significance level |
| What is the size of a test? | Prob. type I error |
| What is the power of a test? | 1- prob. type II error |
| When comparing two tests' powers which is the preferable one? | One with the highest power |
| How do you work out the power function? | Work out probability of falling within the critical region but use p as the probability. |
| How do you sketch power functions? | use suitable integer values |
Created by:
XanderMoore
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