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Probability density function (PDF) is a method to ascertain the random variable’s probability, coming within a range of values, as opposed to taking on any one value. The function elucidates the probability density function of normal distribution and how mean and deviation exists. The standard normal distribution is used in statistics, often used in science to express the real-valued variables whose distribution is unknown.
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Keywords: Probability, Probability density function, Distribution functions, Functions, statistics, graph, integrations
Also read: Isosceles Triangle Theorems
Probability Density Functions
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The Probability Density Function (PDF) describes the probability function expressing the density of a continuous random variable lying between a specific range of values. It is also called the probability distribution function or probability function. But this function is defined in many other sources as the function over a broad set of values. It is mostly considered as the cumulative distribution function and sometimes as the probability mass function (PMF). But, the essentially PDF (probability density function) is defined for continuous random variables, whereas PMF (probability mass function) is defined for discrete random variables
Probability Density Functions Formula
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In a continuous random variable, the probability of X on some given value x is always 0. Here, if we intend to find P (X = x), it will not work. Instead of this, we need to calculate the probability of X lying in an interval (a, b). We have to find it for P (a< X< b), and we can use this formula of PDF. The formula for the probability density function is as follows:
P(a<X<b)= baf(x)dx
Or
P(a≤X≤b) = baf(x)dx
This is because, when X is continuous, we can ignore the endpoints of ranges while finding probabilities of continuous random variables. Which implies, for any constants a and b,
P (a ≤ X ≤ b) = P (a < X ≤ b) = P (a ≤ X < b) = P (a < X < b).
Also read: First Order Differential Equation
Probability Density Function Graph
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Probability density function is an integral of the density of the variable density over a given interval. It is expressed by f (x). This function is either positive or non-negative at any point of the graph, and the integral, more specifically the definite integral of PDF over the entire space is always equal to one. The graph of PDFs characteristically resembles a bell curve, with the probability of the outcomes below the curve. The graph of a probability density function for a continuous random variable x with function f is shown below (x):
Probability Density Function Properties
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The probability density function of a continuous random variable ‘X’ with support S is an integrable function f(x) has the following properties:
- In the support S, f(x) is positive everywhere:
f(x)>0, for all x in S
- In the support S, the area under the curve f(x) is 1:
Sfxdx=1
- If f(x) is the PDF of x, then the integral f(x) across that range gives the probability that belongs to A, where A is some interval:
P(X∈A) = Af(x)dx
Applications of Probability Density Functions
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The probability density function has the following applications:
- The probability density functions are used to model the annual data of atmospheric gasses temporal concentration.
- Probability Density Function is utilized in modeling the diesel engine combustion.
- In Statistics, PDF is applied to calculate the probabilities related to the random variables.
- Probability Distributions are helpful in risk management.
- PDF is also used to assess the probability and amount of loss.
Also read: Definite Integral Formula
Things to Remember
- The probability function which is expressed for the density of a continuous random variable located between a certain range of values is called a Probability Density Function.
- It can also be called a probability distribution function or just a probability function. This function is referred to as a cumulative distribution function or Probability Mass Function (PMF).
- Probability Distribution function (PDF) is for continuous random variables however PMF (Probability Mass Function) is for discrete random variables.
- PMF is not useful for continuous random variables because for any continuous random variable P(X=x) =0 for all x∈R [R = real numbers].
- The function expressed in the form of an integral of the density of the variable density over a given interval is known as PDF and it is generally represented by f (x).
- The function f(x) is either positive or non-negative at any point in the graph.
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Sample Questions
Ques. Let X be a continuous random variable with the PDF given by: f(x)={x; 0<x<1 2-x; 1<x<2 0; x>2 Find P (0.5 < x < 1.5). (5 marks)
Ans: Given PDF
f(x)={x; 0<x<1 2-x; 1,x<2 0; x>2
P(0.5<X<1.5)= \(\int _{0.5}^{1.5}\)f(x)dx
Let us split the integral by taking the intervals as given below:
= \(\int_{0.5}^1\)f(x)dx+\(\int_1^{1.5}\)f(x)dx
Substituting the corresponding values of f(x) based on the intervals, we get;
=\(\int_{0.5}^1\)xdx+\(\int_1^{1.5}\)(2-x)dx
Integrating the functions, we get;
=(\(\frac{x^2}{2}\))0.51+(2x-\(\frac{x^2}{2}\))11.5
= [(1)2/2 – (0.5)2/2] + {[2(1.5) – (1.5)2/2] – [2(1) – (1)2/2]}
= [(½) – (1/8)] + {[3 – (9/8)] – [2 – (½)]}
= (3/8) + [(15/8) – (3/2)]
= (3 + 15 – 12)/8
= 6/8
= 3/4
Ques. What is the probability that X falls between 1/2 and 1? That is, what is P(1/2<X<1)? What is P(X=1/2)? (5 marks)
Ans: It is a straightforward integration to see that the probability is 0:
\(\int_{1/2}^{1/2} 3x^2dx=[x^3]_{x=1/2}^{x=1/2}= \frac{1}{8}- \frac{1}{8}=0\)
In fact, in general, if X is continuous, the probability that X takes on any specific value x is 0. That is, when X is continuous, P(X=x) for all x in the support.
An implication of the fact that P(X=x)=0 for all x when X is continuous is that you can be careless about the endpoints of intervals when finding probabilities of continuous random variables. That is:
P(a≤X≤b)=P(a<X≤b)=P(a≤X<b)=P(a<x<b) for any constants a and b.
Ques. Let X be a continuous random variable with PDF given by fX (x)=1/2 e−|x|, for all x∈ R. If Y=X2, find the CDF of Y. (5 marks)
Ans. First, we note that RY= [0, ∞). For y∈ [0, ∞), we have
Fy (y)= P (Y≤ y)
=P(X2≤y)
= P (−√y≤ X≤√y)
= \(\int\)-√y√y1/2 e−|x| dx
=\(\int\) -√y0√ye−|x| dx
= 1−e−√y.
Thus, FY(y)= 1−e−√y. or 0
Ques; Let X be a random variable with PDF given by fX(x)=cx2 for |x|≤1 OR 0 Find P(X≥1/2) (3 marks)
Ans. To find P(X≥1/2), we can write
P (X≥1/2)
=3/2\(\int_{1/2}^{1}\)x2dx
=7/16.
Ques. Let X be a continuous random variable with PDF fX(x)=4x3 0<x≤1 OR O Find P(X≤\(\frac{2}{3}\)|X>\(\frac{1}{3}\)) (4 marks)
Ans. To find Find P(X≤\(\frac{2}{3}\)|X>\(\frac{1}{3}\))
P(X≤\(\frac{2}{3}\)|X>\(\frac{1}{3}\))=\(\frac{ P( \frac{1}{3}<X \leq \frac{2}{3})}{P(X> \frac{1}{3})}\)
= \(\int_ {\frac{1}{3}}^{\frac{2}{3}}\)4x3dx \ \(\int_ {\frac{1}{3}}^{1}\)4x3dx
= 3/16
Ques. Probability density function of random variable X is given below f(x) = 0.25 if 1≤x≤5 OR 0 p(x≤4)? (5 marks)
Ans. f(x) = 0.25 if 1x≤5 OR 0
Probability density function of random variable X is
\(\int _{- \infty}^{ \infty}\) f(x) dx = 1
P(x≤4) =\(\int _{- \infty}^{4}\)f(x)dx
= \(\int _{- \infty}^{4}\)(0)dx+ \(\int _{1}^{4}\)(0.25)dx
= ¼ (x)41= 1/4 (4-1)
= p (x≤4) = 3/4
Ques. Lex X a random variable with PDF F(x)= 0.2 for |x|≤1; OR 0.1 for 1<|x| ≤4; OR 0 Find the probability P(0.5<X<5). (5 marks)
Ans. Given PDF is
F(x)= 0.2 for |x|≤1; OR 0.1 for 1<|x| ≤4; OR 0
Probability P(0.5<X<5)
P=\(\int _{- \infty}^{ \infty}\)f(x) dx
=\(\int _{0.5}^{1}\)(0.2) dx+ \(\int _{1}^{4}\)(0.1) dx
=0.2 [1 – 0.5] + 0.1 [4 – 1]
P= 0.1× 0.5 × 3
=0.1 + 0.3= 0.4
Ques. If PDF of x is given by f(x)= kx2/2 for -1≤x≤1 OR 0 What is the value of k? (5 marks)
Ans. f(x)= kx2/2 for -1≤x≤1 OR 0
Ques. What is the value of ‘λ' for , f(x)= λ(x-1)(2-x) for 1≤x≤2 OR 0? (5 marks)
Ans. Given, f(x) = λ(x-1) (2-x) for 1≤x≤2 OR 0
Ques. If X is a continuous random variable denoting the temperature measured. The range of temperature is [0, 100] Celsius and probability density function of X is f(x) = 0.01 for 0≤ X ≤ 100. What is mean of X? (3 marks)
Ans. The Probability density function of X is f(x) = 0.01 for 0≤ X ≤100
The mean or expectation of a continuous random variable is given by
E(x) = \(\int_0^{100}\) x (0.01) dx = 50
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