Continuous Distributions

Natural random variables have sets of uncountable possible values.

Continuous Uniform Distribution

Continuous Uniform Distribution

A random variable is said to follow a uniform distribution over the interval if the probability density function is given by:

Then, is denoted .

It can be shown then, that


The probability density function looks like the above graph, where every point has the same probability .

Exponential Distribution

Exponential Distribution

A continuous random variable is said to follow an exponential distribution with parameter if probability density function is given by:

It can then be shown that

Alternative form

where

The cumulative distribution function can be derived using the integration of the probability density function:

for .

# calculating exponential probability
 
dexp(x, lambda) # f(x)
pexp(x, lambda) # P(X <= x)
pexp(x, lambda, lower.tail = F) # P(X > x)

Theorem 15

Suppose that has an exponential distribution with .

For any two positive numbers ,

The interpretation of the above theorem is that the exponential distribution is “memoryless”. It means that given two values , is the same as the conditional probability of a conditional probability with the same offset .


Example

Assume the life length of a bulb follows the exponential distribution; ,

If the bulb has lasted for hours, the probability that it will last for another hour (given the previous information) is the same as the probability it will last for hour given it is brand new.

Normal Distribution

Normal distribution

A random variable is said to follow a normal distribution with parameters if its probability density function is given by:

It is then denoted

It can be then shown that:

The probably density function of the normal distribution is

• positive over the whole real line
• symmetrical about
• bell-shaped

Properties

1. The total area under the curve above the horizontal axis is equal to .

• This validates that is a probability density function.

2. Normal curves are identical in shape if the is the same, but are centered at different points if the is different.
3. As increases, the curve flattens. (and vice versa)
4. Given that ,

then follows the distribution.

Z has a standard normal distribution, and the probability density function is given:

Calculating normal probabilities

Difficult as there is no close formula, and computation relies on numerical integration (since it is a continuous variable).

Thus to calculate probabilities:

1. Transform the distribution into a standard normal distribution (i.e., Property 4)
2. Use the standard normal to calculate the probability

dnorm(x, mu, sigma) #f(x);
pnorm(x, mu, sigma) #F(x) = P(X <= x)
pnorm(x, mu, sigma, lower.tail = F) # P(X > x)

Quantile

The quantile where of a random variable is the number that satisfies

Z upper quantile

denotes the upper quantile of .

Common values include:

As the PDF of is symmetrical about ,

# computing normal quantiles
qnorm(alpha, mu, sigma, lower.tail = F) # x_(alpha)
qnorm(alpha, mu, sigma) # x_(1-alpha)
qnorm(alpha, lower.tail = F) #z_-_alpha
qnorm(alpha) #z_(1-alpha)

Approximation of Binomial

Rule of thumb

The normal approximation can be done as follows: given , as ,

Continuity correction

The continuity correction factor accounts for the fact that a normal distribution is continuous, and a binomial is not. Generally, it just subtracts or adds to the value.