Distribution Function Definition

Let X be an random variable, then the function such that F:R→R defined by F(X) = P(X ≤ x) is called distribution function of random variable X.

Note 1: F(-∞) = 0 and F(∞) = 1

Note 2: If F(X) is a distribution function of a random variable X if a < b, then P(a < x ≤ b) = F(b) – F(a)

Note 3: P(a ≤ x ≤ b) = P(x = a) + F(b) – F(a)

Distribution function of discrete random variables

Definition

Let X be a random variable whose space is {x₁, x₂, x₃, … , x_n}. Let P( X = x_i) = p_i then the distribution function F:R→R of the discrete random variable X is given by
\[\text{F}\left( \text{X} \right)\text{ }=\text{ P}(\text{X}\le \text{x})\text{ }=\sum\limits_{{{x}_{i}}\le x}{{{p}_{i}}}\]
and F(X) is called the Probability Mass Function (PMF) of the discrete random variable X.

Note: Let P( X = x_i) = p_i then the definition of probability, it follows that Σ p_i = 1.

Distribution function of continuous random variables

Definition

Let X be a continuous random variable with probability density function f(x), we define F:R→R is given by
\[F(x)=\int\limits_{-\infty }^{x}{f(x)dx}\]
is called the distribution function of the continuous random variable X.

Note: From the definition of F(x) we have the following property,
\[(i)F(-\infty )=0\]
\[(ii)F(\infty )=1\]
\[(iii)P(a\le x\le b)=F(b)-F(a)=\int\limits_{a}^{b}{f(x)dx}\]
\[=\int\limits_{-\infty }^{b}{f(x)dx-}\int\limits_{-\infty }^{a}{f(x)dx}\]

Verify that p(x) defined by

Find the probability that the variate having this density will fall in the interval [1.5, 2.5]. Also, evaluate the cumulative distribution function F(2.5).

Solution:

Evidently, the given p(x) ≥ 0. Further,
\[P(-\infty <x<\infty )=\int\limits_{-\infty }^{\infty }{p(x)dx}\]
\[\Rightarrow P(-\infty <x<\infty )=\int\limits_{-\infty }^{0}{p(x)dx}+\int\limits_{0}^{\infty }{p(x)dx}\]
\[\Rightarrow P(-\infty <x<\infty )=0+\int\limits_{0}^{\infty }{{{e}^{-x}}dx}\]
\[\therefore P(-\infty <x<\infty )=1\]
The two conditions verified above show that p(x) is a probability density function.

Next, we find that
\[P(1.5<x<2.5)=\int\limits_{1.5}^{2.5}{p(x)dx}\]
\[\Rightarrow P(1.5<x<2.5)=\int\limits_{1.5}^{2.5}{{{e}^{-x}}dx}\]
\[\therefore P(-\infty <x<\infty )=-\left( {{e}^{-2.5}}-{{e}^{-1.5}} \right)=0.141\]
Lastly we find that
\[P(2.5)=\int\limits_{-\infty }^{2.5}{p(x)dx=\int\limits_{0}^{2.5}{{{e}^{-x}}dx}}\]
\[\Rightarrow P(2.5)=-\left( {{e}^{-2.5}}-1 \right)=0.9179\]

For the distribution given by the cumulative function

Find density function. Also, evaluate
\[(i)P(0.5<X<0.75)\]
\[(ii)P(X\le 0.5)\]
\[(iii)P(X>0.75)\]
Solution:

We know that, If F(x) is the cumulative distribution function (cdf) the probability density function f(x) can be obtained by
\[f(x)=\frac{d}{dx}F(x)=F'(x)\]
Given, cumulative distributive function


Also,
\[P(0.5<X<0.75)=F(0.75)-F(0.5)\]
\[\Rightarrow P(0.5<X<0.75)={{\left( 0.75 \right)}^{2}}-{{\left( 0.5 \right)}^{2}},\left[ \because F(t)=\int{f(x)dx} \right]\]
\[\therefore P(0.5<X<0.75)=0.3125\]
\[Now,P(X\le 0.5)=F(0.5)={{\left( 0.5 \right)}^{2}}=0.25\]
\[Again,P(X>0.75)=1-P(X\le 0.75)=1-F(0.75)\]
\[\therefore P(X>0.75)=1-{{\left( 0.75 \right)}^{2}}=0.4325\]

The pdf of a random variable X is given by

Find (i) cumulative distribution function and (ii) P(X ≥ 1.5)

Solution:

By using formula
\[F(t)=P(X\le t)=\int\limits_{-\infty }^{t}{f(x)dx}\]

For -∞ < t < 0, we get

\[F(t)=\int\limits_{-\infty }^{0}{f(x)dx=}\int\limits_{-\infty }^{0}{0dx=}0\]
For 0 ≤ t ≤ 1, we get
\[F(t)=\int\limits_{-\infty }^{0}{f(x)dx+\int\limits_{0}^{t}{f(x)dx}}\]
\[\Rightarrow F(t)=0+\int\limits_{0}^{t}{xdx}=\frac{{{t}^{2}}}{2}\]
For 1 < t ≤ 2, we get
\[F(t)=\int\limits_{-\infty }^{0}{f(x)dx+\int\limits_{0}^{1}{f(x)dx}}+\int\limits_{1}^{t}{f(x)dx}\]
\[\Rightarrow F(t)=\int\limits_{-\infty }^{0}{0dx+\int\limits_{0}^{1}{xdx}}+\int\limits_{1}^{t}{\left( 2-x \right)dx}\]
\[\therefore F(t)=1-\frac{1}{2}{{\left( t-2 \right)}^{2}}\]
For t > 2, we get
\[F(t)=\int\limits_{-\infty }^{0}{f(x)dx+\int\limits_{0}^{1}{f(x)dx}}+\int\limits_{1}^{2}{f(x)dx}+\int\limits_{2}^{t}{f(x)dx}\]
\[\Rightarrow F(t)=\int\limits_{-\infty }^{0}{0dx+\int\limits_{0}^{1}{xdx}}+\int\limits_{1}^{2}{\left( 2-x \right)dx}+\int\limits_{2}^{t}{0dx}\]
\[\therefore F(t)=\frac{1}{2}+\left( 2-\frac{3}{2} \right)=1\]
Thus the required cdf is

\[Now,P(X\ge 1.5)=\int\limits_{1.5}^{\infty }{f(x)dx}\]
\[\therefore P(X\ge 1.5)=\int\limits_{1.5}^{2}{\left( 2-x \right)dx}+\int\limits_{2}^{\infty }{0dx=\frac{1}{8}}\]

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