Probability and Statistics Lecture 7

Continued

Probability and Statistics Lecture 6

Random Variables and Probability Distribution

Random Variable and its Probability Distributions

Random Variables

Imagine we are taking into account the values that a die can give us. One can call the random variable as $X$ and the possible values it can take as $x$.

• $X$ can take on the values $1,2,3,4,5,6$.
• It can be represented as $X=x$ for $x\in 1,2,3,4,5,6$. $X$ here is the random variable

Definition

A random variable is a function that assigns a numerical value to each sample point in a sample set (sample space) of a random phenomenon. It can be classified as:

1. Discrete Random Variables: Take on a finite or countable set of values (e.g., number of defective items in a batch).
2. Continuous Random Variables: Take on an infinite set of values within a range (e.g., the time it takes for a radioactive particle to decay).

Example: Two Coins Tossed

Outcome (S)	HH	TH	HT	TT
X (Values)	2	1	1	0

• Sample Space (S): The set of all possible outcomes.
• Sample Point: An individual outcome in the sample space.

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Probability Distribution

In the above example we had $X$ taking on values $1,2,3,4,5,6$. We can assign probabilities to these values with factual information. This is what a probability distribution does. So for each value of $x$ on $X$ we have a probability $P(X=x)$. This is called the probability mass function (PMF).

Definition

A probability distribution describes how probabilities are assigned to possible values of a random variable. It determines the likelihood of events.

Types:

1. For Discrete Random Variables:
   • Represented by a probability mass function (PMF).
   • Example: Rolling a die has $P(X=x)=16P(X=x)=\frac{1}{6}forx\in 1,2,3,4,5,6x\in 1,2,3,4,5,6$.
2. For Continuous Random Variables:
   • Represented by a probability density function (PDF).
     • Meaning there is a function $f(X=x)$ such that for any value x it has a defined output.
   • Probabilities are calculated as areas under the curve of the PDF.

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Random Variable Types

• Discrete Random Variable: Has finite or countable values.
  • Example : Number of telephone calls received per unit time in a telephone exchange.
  • Example : Number of second year students who have got about 90% marks
  • Example : Number of alpha particles by a radioactive substance.
• Continuous Random Variable: Has infinite values. Its Uncountable

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Need for This Method

This method helps calculate probabilities in situations such as tossing a coin until a head appears or analysing decay time in nuclear processes.

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Example in Nuclear Problems

Random variables can describe phenomena such as:

1. Decay Time: Time taken for a radioactive atom to decay (continuous random variable).
2. Number of Neutrons Released: In a nuclear reaction (discrete random variable). Steps:
3. Assign a real number to the decay time of each atom (a random sample point).
4. Analyse the probability distribution of decay times to predict the system’s behavior.

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Map

• To solve questions one can refer to
  • Probability and Statistics Lecture 7 -
• R studio
  • Probability and Statistics Lab 4 Part 1
  • Probability and Statistics Lab 4 Part 2.

References

• Probability and Statistics Lectures at MPSTME

Information

• date: 2025.03.13
• time: 07:50
• Continued to

Link to original

Question

Question 1

Suppose two fair die are rolled & The sum of the die is noted as a random variable. Obtain the probability distribution of $X$

Answer 1

Sample space of two die rolled $n(s)=36$ $X:Sumondice$

X	2	3	4	5	6	7	8	9	10	11	12
$P(X=x_i)$	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36
Checking if

• $0\le P_i\le 1,\forall =1,2,3,...,11$
• ${\sum }_{\infty }^{i=1}{P}_i=1$

$\therefore$ $P_i$‘s probability mass function and $\{x_i,p_i\}$ is probability distribution

References

Continued

Probability and Statistics Lecture 8