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Probability

Random Variable and Its Probability Distribution

Probability Distribution: One day it just comes to your mind to count the number of cars passing through your house. It is not pre-decided that which color car will first pass. This activity or experiment is random. Here we are not interested in the type of cars passing but in their number. Also, we further note down the number of different colored cars passing.

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The number of these cars can be anything starting from zero but it will be finite. This is the basic concept of random variables and its probability distribution. Here the random variable is the number of the cars passing. It is not constant. It can also vary from the type of the events we are interested in.

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Random Variables

A variable is something which can change its value. It may vary with different outcomes of an experiment. If the value of a variable depends upon the outcome of a random experiment it is a random variable. A random variable can take up any real value.

Mathematically, a random variable is a real-valued function whose domain is a sample space S of a random experiment. A random variable is always denoted by capital letter like X, Y, M etc. The lowercase letters like x, y, z, m etc. represent the value of the random variable.

Consider the random experiment of tossing a coin 20 times. You will earn Rs. 5 is you get head and will lose Rs. 5 if it a tail. You and your friend are all set to see who will win the game by earning more money. Here, we see that the value of getting head for the coin tossed for 20 times is anything from zero to twenty. If we denote the number of a head by X, then

X = {0, 1, 2, … , 20}. The probability of getting a head is always ½.

Properties of a Random Variable

  • It only takes the real value.
  • If X is a random variable and C is a constant, then CX is also a random variable.
  • If X1 and X2 are two random variables, then X1 + X2 and X1 X2 are also random.
  • For any constants C1 and C2, C1X1 + C2X2 is also random.
  • |X| is a random variable.

Types of Random Variable

A random variable can be categorized into two types.

Discrete Random Variable

As the name suggests, this variable is not connected or continuous. A variable which can only assume a countable number of real values i.e., the value of the discrete random sample is discrete in nature. The value of the random variable depends on chance. In other words, a real-valued function defined on a discrete sample space is a discrete random variable.

The number of calls a person gets in a day, the number of items sold by a company, the number of items manufactured, number of accidents, number of gifts received on birthday etc. are some of the discrete random variables.

Continuous Random variable

A variable which assumes infinite values of the sample space is a continuous random variable. It can take all possible values between certain limits. It can also take integral as well as fractional values. The height, weight, age of a person, the distance between two cities etc. are some of the continuous random variables.

Probability Distribution

For any event of a random experiment, we can find its corresponding probability. For different values of the random variable, we can find its respective probability. The values of random variables along with the corresponding probabilities are the probability distribution of the random variable.

Assume X is a random variable. A function P(X) is the probability distribution of X. Any function F defined for all real x by F(x) = P(X ≤ x) is called the distribution function of the random variable X.

Properties of Probability Distribution

  • The probability distribution of a random variable X is P(X = xi) = pi for x = xi and P(X = xi) = 0 for x ≠ xi.
  • The range of probability distribution for all possible values of a random variable is from 0 to 1, i.e., 0 ≤ p(x) ≤ 1.

Probability Distribution of a Discrete Random Variable

If X is a discrete random variable with discrete values x1, x2, … , xn, … then the probability function is P(x) = pX(x). The distribution function is

FX(x) = P(X ≤ xi) = ∑i p(xi) = pi

if x = xi and is 0 for other values of x. Here, i = 1, 2, … , n, …

Consider an example of tossing of two fair coins. The possible outcomes for this random experiment are S = {HH, HT, TH, TT}. If X is a random variable for the occurrence of the tail, the possible values for X are 0, 1, and 2. The distribution function for X is F(x) = P(X ≤ x) is

Value of X 0 1 2
P(X = x) = p(x) 1⁄4 2⁄4 1⁄4
F(X) = P(X ≤ x)  = ∑i p(xi) 1⁄4 3⁄4 4⁄4 = 1

Probability Distribution of a Continuous Random Variable

If X is a discrete random variable with discrete values x1, x2, … , xn, … then the probability distribution function is F(x) = pX(xi). The distribution function for a continuoous random variable is

FX(x) = ∫pX(xi) dx

where i = 1, 2, … , n, …

Solved Example for You

Question 1: Three fair coins are tossed. Let X = the number of heads, Y = the number of head runs. (A ‘head run’ is a consecutive occurrence of at least two heads.) Find the probability function of X and Y.

Answer : The possible outcomes of the experiment is S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. X is the number of heads. It takes up the values 0, 1, 2, and 3.

probability distribution

  • P(no head) = p(0) = 1⁄8
  • P(one head) = p(1) = 3⁄8
  • P(two heads) = p(2) = 3⁄8
  • P(three heads) = p(3) = 1⁄8
Value of X, x 0 1 2 3
p(x) 1⁄8 3⁄8 3⁄8 1⁄8

Y is the number of head runs. It takes up the values 0 and 1.

P(Y = 0) = p(0) = 5⁄8, and P(Y = 1) = p(1) = 3⁄8.

Value of Y, y 0 1
p(y) 5⁄8 3⁄8

Question 2: What is a random variable?

Answer: A variable refers to anything which is able to change its value. It can vary with different results of an experiment. If the value of a variable is dependent on the result of a random experiment, it refers to a random variable. Moreover, a random variable may take up any real value.

Question 3: What are the properties of a random variable?

Answer: A random variable merely takes the real value. For instance, if X is a random variable and C is a constant, then CX will also be a random variable. If X1 and X2 are 2 random variables, then X1+X2 plus X1 X2 will also be random. Further, for any constants C1 and C2, C1X1 + C2X2 will also be random. Thus, X will be a random variable.

Question 4: What do you mean by probability distribution?

Answer: We can find the corresponding probability for any event of a random experiment. Similarly, for different values of the random variable, one can find the respective probability of it. Further, the values of random variables along with the corresponding probabilities refer to the probability distribution of the random variable.

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