ZIMSEC A Level · Statistics · Form 5, Form 6
Discrete random variables
A probability distribution lists values x with probabilities summing to 1.
E(X) = Σx P(X = x); Var(X) = Σx² P(X = x) − [E(X)]². E(aX + b) = aE(X) + b; Var(aX + b) = a²Var(X).
The binomial distribution, X ~ B(n, p)
Conditions: fixed number of independent trials, two outcomes, constant probability p.
P(X = r) = ⁿCᵣ pʳ(1 − p)ⁿ⁻ʳ; mean np; variance np(1 − p).
Worked example: 30% of seeds fail to germinate. In a packet of 10, find P(exactly 2 fail).
P(X = 2) = ¹⁰C₂ × 0.3² × 0.7⁸ = 45 × 0.09 × 0.05765 = 0.2335
The Poisson distribution, X ~ Po(λ)
Models random events in a fixed interval of time or space, occurring independently at a constant average rate λ.
P(X = r) = e^(−λ) λʳ/r!; mean = variance = λ.
Worked example: A clinic receives an average of 3 emergency calls per hour. Find P(no calls in an hour) and P(at most 1 call).
P(X = 0) = e⁻³ = 0.0498
P(X ≤ 1) = e⁻³(1 + 3) = 0.199
The Poisson can approximate B(n, p) when n is large and p is small (λ = np).
Check yourself
- X ~ B(8, 0.5). Find E(X) and Var(X).
- X ~ Po(2). Find P(X = 2).
- State two conditions for a binomial distribution.
Show answers
- E(X) = 4; Var(X) = 2
- e⁻² × 4/2 = 0.271
- Any two: fixed number of trials, independent trials, two outcomes, constant probability of success.
Questions about this topic (0)
No questions about this topic yet.
Sign in or create a free account to join in.