Topic 5 - Continuous Random Variables

Question 1

What is a continuous random variable

Answer 1

• ## A variable that can take any value in an interval

Question 2

Why are probability distribution functions irrelevant for c.r.v and what do we use instead

Answer 2

• For c.r.v the probability of taking a specific value is 0
• Therefore we use probability density functions for c.r.v

Question 3

What are the properties of a probability density function

Answer 3

• 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

Question 4

How can we calculate P(a < X < b)

Answer 4

• P(a < X < b) = F(b) - F(a)

Question 5

What is the difference in cumulative distribution functions for discrete and continuous r.v

Answer 5

• Discrete CDF rise in jumps
• Continuous CDF rise smoothly

Question 6

What is the uniform distribution

Answer 6

• A probability distribution that has equal probabilities for all possible outcomes of the random variable

Question 7

What is E(X) and Var(X) of a uniform distribution

Answer 7

• E(X) = a + b /2
• Var(X) = (b-a)^2 / 12
• Where values lie in the interval a <= x <= b

Question 8

What is the central limit theorm

Answer 8

• 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

Question 9

What are the 5 characteristics of the normal distribution

Answer 9

• 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

Question 10

How does changes in S.D influence how the normal distribution looks

Answer 10

• Larger values in S.D will result in wider, flatter curves

Question 11

What is the probability density function for a normal distribution

Answer 11

• f(x) = (1/sqrt 2 * pi * Var(X)) * e^-(x-mu)^2 / 2 * Var(X)

Question 12

What are the properties of the standardised normal distribution

Answer 12

• Mean = 0, Variance = 1

Question 13

How do we turn X ~ N(μ, σ^2) into a standardised version of X

Answer 13

• Z = X - μ / σ

Question 14

How is a normal distribution denoted

Answer 14

• X ~ N(μ, σ^2)

Question 15

Why would we want to convert X into Z for finding probabilites

Answer 15

• 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

Question 16

If our value of Z is negative, how do we calculate its probability

Answer 16

• Use the fact that the curve is symetric
• e.g. P(Z < -2.00) = 1 - P(Z < 2.00)

Question 17

What formula would we use if we were trying to figure out our X value

Answer 17

• X = μ + Zσ

Question 18

When would we use a normal distribution to approximate a binomial

Answer 18

• When n is large and p is around 0.5
• or when np(1-p) > 9

Question 19

If we are approximating a binomial from a normal how do we calculate Z

Answer 19

• Z = X - np / sqrt np(1-p)

Question 20

What is the exponential distribution used to model

Answer 20

• The length of time between two occurances of an event

Question 21

When is a c.r.v T(t>0) said to have an exponential distribution with parameter λ

Answer 21

• If its probability density function is as follows
• f(t) = λe^-λt for t>0

Question 22

What is the cumulative distribution function for an exponential distribution

Answer 22

• F(t) = P(T < t) = 1 - e^-λt

Question 23

What special property does the exponential distribution have

Answer 23

• 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

Question 24

What is the poisson process

Answer 24

• 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