Random Variables and Distributions
This section of CS1 tests whether you can specify a random variable's distribution, move between pmf/pdf and cdf, and compute moments without relying on a formula sheet alone. You should recognise the standard families, their parameterisations and the situations each models.
Core rules
- For discrete X, p(x)=P(X=x) with sum 1; for continuous X, f(x) is a density and P(a<X<b)=∫_a^b f(x)dx. A density value is not a probability and may exceed 1.
- The cdf F(x)=P(X≤x) is non-decreasing, right-continuous, tends to 0 and 1, and differentiates to f where continuous.
- E[g(X)] = Σ g(x)p(x) or ∫ g(x)f(x)dx; Var(X)=E[X²]−E[X]².
- Know binomial, Poisson, geometric, negative binomial, exponential, gamma, normal, uniform and beta, including mean, variance and when they arise.
Worked example
Let X~Binomial(n,p). Write X=Σ_{i=1}^n I_i, where I_i is the indicator that trial i succeeds. Then E[I_i]=p and E[X]=ΣE[I_i]=np. For the variance, independence gives Var(X)=ΣVar(I_i)=Σ(p−p²)=np(1−p). The indicator method avoids evaluating a sum and is accepted in CS1.
Traps
- Exponential/gamma parameterisations: rate λ versus scale θ=1/λ. Check which one the question uses.
- Ignoring support: a density such as f(x)=2x on (0,1) is zero outside that interval.
- Using f(x) as P(X=x), or forgetting the Jacobian when transforming.
- Quoting a mean/variance for the wrong family, e.g. confusing geometric counts (trials vs failures).