1.6. Density and Mass Functions

(1)

1.6. Density and Mass Functions

Definition 1.6.1 (Probability Mass Function)

The probability mass function (pmf) of a discrete random variable X is given by fX(x) = P (X = x) for all x.

Example 1.6.2 (Geometric probabilities)

For the geometric distribution of Example 1.5.4, we have the pmf

f_X(x) = P (X = x) =







(1 − p)^x−1p for x = 1, 2, . . .

0 otherwise.

From this example, we see that a pmf gives us “point probability”. In the discrete case, we can sum over values of the pmf to get the cdf. The analogous procedure in the continuous case is to substitute integrals for sums, and we get

P (X ≤ x) = F_X(x) = Z _x

−∞

f_X(t)dt.

Using the Fundamental Theorem of Calculus, if f_X(x) is continuous, we have the further relationship

d

dxF_X(x) = f_X(x).

Definition 1.6.3 (Probability Density Function)

The probability density function or pdf, f_X(x), of a continuous random variable X is the function that satisfies

F_X(x) = Z _x

−∞

f_X(t)dt for all x. (1)

A note on notation : The expression “X has a distribution given by F_X(x)” is abbreviated symbolically by “X ∼ F_X(x)”, where we read the symbol “∼” as “is distributed as”. We can similarly write X ∼ f_X(x) or, if X and Y have the same distribution, X ∼ Y .

1

(2)

In the continuous case we can be somewhat cavalier about the specification of interval probabilities. Since P (X = x) = 0 if X is a continuous random variable,

P (a < X < b) = P (a < X ≤ b) = P (a ≤ X < b) = P (a ≤ X ≤ b).

Example 1.6.4 (Logistic probabilities) For the logistic distribution, we have

F_X(x) = 1 1 + e^−x, hence, we have

f_X(x) = d

dxF_X(x) = e^−x (1 + e^−x)², and

P (a < X < b) = F_X(b) − F_X(a)

= Z _b

−∞

f_X(x)dx − Z _a

−∞

f_X(x)dx

= Z _b

a

f_X(x)dx.

Theorem 1.6.5

A function FX(x) is a pdf (or pmf) of a random variable X if and only if a. FX(x) ≥ 0 for all x.

b. P

xfX(x) = 1 (pmf) or R_∞

−∞fX(x)dx = 1 (pdf).

Proof: If f_X(x) is a pdf (or pmf), then the two properties are immediate from the defini- tions. In particular, for a pdf, using (1) and Theorem 1.5.3, we have that

1 = lim

x→∞F_X(x) = Z _∞

−∞

f_X(t)dt.

The converse implication is equally easy to prove. Once we have f_X(x), we can define F_X(x) and appeal to Theorem 1.5.3. Theorem 1.5.3 is the Theorem in Lecture note 5 on cumulative distribution function. ¤

2