X - B(n,p) where n = Number of Trials, p = Probability of Success
P(X=r) = nCr x pr x (1-p)n-r
Mean: E(X) = np
Variance: Var(X) = np(1-p)
Standard Deviation: √[np(1-p)]
Criteria for using the Binomial Distribution:
- There must be a fixed number of trials
- Each trial must have the same two possible outcomes
- The trials must be independent on each other
- The probability of success must be the same for each trial
X - Po(λ)
- The Poisson Distribution is a probability distribution.
- It is used when a random event occurs at a constant average rate
- It gives the probability for every number of events in a fixed period of time or area of space
P(X=r) = (e^-λ x λr ) / r!
Criteria for using the Poisson Distribution:
- Events must occur independently of one another
- They must happen one at a time
Finding the Mode of a Poisson Distribution:
- If λ is a whole number then the distribution is bimodal. The modes are λ-1 and λ.
- If λ is not a whole number then the distribution is unimodal. The mode is the nearest whole number less than λ.
Mean = E(X) = λ
Variance = Var(X) = λ
Using the Poisson Distribution to approximate the Binomial Distribution
X - B(n,p). When n is a very large value and p is close to zero, the binomial can be distributed with X - Po( λ) where λ = np.
Continuous Random Variables - PDF's
For a Probability Density Function:
- The probability that X is between a and b is the area under the curve between a and b: P(a≤X≤b) = ∫ f(x) dx with limits a and b.
- The probability density cannot be negative
- The total area under the curve = 1
∫ f(x) dx = 1
You need to be able to:
1. Recognise a PDF
- Are any probability densities negative?
- Is the area under the graph equal to 1?
2. Calculate probabilities
3. Define a PDF
Continuous Random Variables - CDF's
For a Cumulative Density Function, F(x) = P( X≤x) ∫ f(x) x with limits X and ∞ With Cumulative Distribution Functions it is necessary to be able to:
- Calculate probabilities
- Convert PDF's to CDF's (integration)
- Convert CDF's to PDF's (differentiation)
Mean and Variance of Continuous Random Variables
Expectation and Variance of a Continuous Random Variable:
Expected Value = ∫ x f(x) dx over all possible values of x
Variance = ∫ x² f(x) dx - E(x) over all possible values of x