Topic 5 - Continuous Random Variables Flashcards
What is a continuous random variable
- ## A variable that can take any value in an interval
Why are probability distribution functions irrelevant for c.r.v and what do we use instead
- For c.r.v the probability of taking a specific value is 0
- Therefore we use probability density functions for c.r.v
What are the properties of a probability density function
- f(x) > 0 for all x
- The whole area under the curve is equal to 1
- The probability that X is between two values is the area under the curve between those two values
- The cumulative distribution function F(x0) is the area under the probability density function from min x to x0
How can we calculate P(a < X < b)
- P(a < X < b) = F(b) - F(a)
What is the difference in cumulative distribution functions for discrete and continuous r.v
- Discrete CDF rise in jumps
- Continuous CDF rise smoothly
What is the uniform distribution
- A probability distribution that has equal probabilities for all possible outcomes of the random variable
What is E(X) and Var(X) of a uniform distribution
- E(X) = a + b /2
- Var(X) = (b-a)^2 / 12
- Where values lie in the interval a <= x <= b
What is the central limit theorm
- States that under very weak assumptions, the sum of a large number of independent and identically distributed random variables has an approximate Normal Distribution, regardless of the distribution of the individual r.v.s
What are the 5 characteristics of the normal distribution
- Bell shaped and symetrical
- Mean, mode and median are equal
- Location is determined by the mean
- Spread is determined by S.D
- Has a theoretical infinite range
How does changes in S.D influence how the normal distribution looks
- Larger values in S.D will result in wider, flatter curves
What is the probability density function for a normal distribution
- f(x) = (1/sqrt 2 * pi * Var(X)) * e^-(x-mu)^2 / 2 * Var(X)
What are the properties of the standardised normal distribution
- Mean = 0, Variance = 1
How do we turn X ~ N(μ, σ^2) into a standardised version of X
- Z = X - μ / σ
How is a normal distribution denoted
- X ~ N(μ, σ^2)
Why would we want to convert X into Z for finding probabilites
- Finding probabilites for values of X is very difficult as we are not given probabilites, if we convert to Z we can use the standard normalised table to find probabilites
If our value of Z is negative, how do we calculate its probability
- Use the fact that the curve is symetric
- e.g. P(Z < -2.00) = 1 - P(Z < 2.00)
What formula would we use if we were trying to figure out our X value
- X = μ + Zσ
When would we use a normal distribution to approximate a binomial
- When n is large and p is around 0.5
- or when np(1-p) > 9
If we are approximating a binomial from a normal how do we calculate Z
- Z = X - np / sqrt np(1-p)
What is the exponential distribution used to model
- The length of time between two occurances of an event
When is a c.r.v T(t>0) said to have an exponential distribution with parameter λ
- If its probability density function is as follows
- f(t) = λe^-λt for t>0
What is the cumulative distribution function for an exponential distribution
- F(t) = P(T < t) = 1 - e^-λt
What special property does the exponential distribution have
- The memoryless property that states that an arrival does not affect the probability of waiting time untill the next arrival
- e.g. if you’ve waited hours for a success, the success isn’t any more likely to come
What is the poisson process
- If waiting times between events are independently exponentially distributed with parameter λ, then the number of events in one unit of time has a poisson distribution
