Download Discrete Distributions

Continuous Distributions
Discrete Distributions place probability on specific numbers. For example, the Binomial distribution
places probability only on the values 0,1,2, …, n. This is why the probability distributions for discrete
random variables are often referred to as probability mass functions.
Some random variables, however, can take on all values in a certain range. For example, a person’s
height can be anywhere between 0 and 10 feet tall. It might be exactly 6 feet, but odds are it’s more like
6.00234556 feet. Note: if you have an interval of numbers, you can’t place a positive probability on each
of them. Therefore, we cannot talk about the probability that a certain person is exactly 6 feet. However,
we can still talk about the probability of a randomly chosen person being between 71.8 and 72.2 inches.
To do so, we need another way of expressing probability models for random variables that can take on
any value in an interval of the real line. Such random variables are called continuous random variables.
Just like discrete random variables, continuous random variables have a range of possible values.
However, this range consists of an interval of numbers (xmin, xmax) instead of a set of numbers like
0,1,2,.., n. To compute probabilities for continuous random variables, we have a probability density
function, f(x), defined over the range of possible values.
Suppose we wish to find the probability that a continuous random variable is between a and b. The
process is similar to how we would find the probability that a discrete random variable is between a and
b. For a discrete random variable, we would add the probabilities for the possible values the random
variable could take on between a and b. In the case of a continuous random variable, we integrate the
density over the range. Therefore integration replaces summations.
Not just any f(x) can be a density. Since integrating densities is supposed to give us probabilities,
densities must obey certain conditions.
x max
1.
 f x dx = 1. Just like any other prob. distribution, the probability of the sample space must be 1.
x min
2. f(x) ≥ 0 for all x.
If f(x) were negative anywhere, we could integrate in that region and get a negative
probability. Since probabilities cannot be negative, densities can’t be either.
One important fact to keep in mind when dealing with densities is that all individual points have
probability 0. Thus, if X is a continuous random variable, P(X = d) = 0 for any d. Points do not have
probability, but intervals do have probability.
Example: Let X be a random variable with density f(x) = 2x over the range 0 < x < 1.
1. Check that this is a valid probability density.
2. Find P(0.2 < X < 0.6). Answer: 0.32
Notice that since points don’t have probabilities,
P(0.2 < X < 0.6) = P(0.2 ≤ X ≤ 0.6) = P(0.2 ≤ X < 0.6) = P(0.2 < X ≤ 0.6)
Also, be careful to keep the range in mind when computing probabilities. Continuing with the example
above, suppose we wanted to compute P(-0.4 < X < -0.1). There is no need to do the integration because
the values are outside the range of possible values for X. Also, if we wanted to compute
P(-0.3 < X < 0.7), we would really need to compute P(0 < X < 0.7). In general, look at the probability you
are interested in, decide where you need to integrate based on the range of the random variable, and only
then perform the integration to find the probabilities.
Cumulative Distribution Functions
We are interested in interval probabilities such as P(a < X < b). Instead of doing the integration again and
again for every a and b , one useful function is the cumulative distribution function, or cdf. The
notation is F(x) = P(X ≤ x). F(x) is defined by
0 x  x min
 x

P  X  x    f t dt x   x min , x max 
x

1 x  x max

min
Notice that we switched the variable of integration from x to t. The variable of integration is irrelevant as
long as it does not match the letter used for the boundary of integration. Thus, for the cdf, the boundary
of integration involves x and we use t here for the variable of integration.
For an example of a cdf, consider the density f(x) = 2x for x between 0 and 1. The cdf in that range is
x
F(x) =
 2t dt  t
2 x
0
 x2
0
The cdf allows us to compute many kinds of probabilities directly, without performing any integration.
We perform the integration once to find the cdf and then use that function.
1. P(X ≤ x) = F(x)
2. P(X > x) = 1 – P(X ≤ x) = 1 – F(x)
3. P(a < X ≤ b) = P(X ≤ b) – P(X ≤ a) = F(b) – F(a)
Again note that for continuous random variables, points have no probability.
Expected Values and Variances
Recall that, for discrete random variables, we had formulas for
E(X) = ∑ x P(X=x) = µX
E[h(X)] = ∑ h(X) P(X=x)
V(X) = σX 2 = E[ (X – μ)2] = Σ (x – μ)2 P(X = x) = E(X 2) – (E[X])2
Similar formulas hold for continuous distributions, except that the sums are replaced by integrals and the
P(X = x) is replaced with the density f(x).
x max
E(X) =
x max
 x f x dx
In general, E[h(X)] =
x min
x min
x max
V(X) = E(X 2) – (E[X])2 =
 hx  f x dx

x min
x



2
x f  x  dx –  x f  x  dx 
x



2
max

min
In our example, f(x) = 2x for 0 < x < 1. Then E(X) = 2/3; E(X 2) = 1/2; V(X) = 1/18
Note: The standard deviation of X is still the square root of V(X).