# Ch 4.1 Discrete Random Variable

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Ch 4.1 Discrete Random Variable

### Random Variable

Random Variable: is a variable (X) that has a single numerical value, determined by chance, for each outcome of a procedure.

Probability distribution: is a table, formula or graph that gives the probability of each value of the random variable.

A Discrete Random Variable has a collection of values that is finite or countable similar to discrete data.

A Continuous Random Variable as infinitely many values and the collection of values is not countable.

Ex : X = the number of times “four” shows up after tossing a die 10 times is a discrete random variable.

X = weight of a student randomly selected from a class. X is a continuous random variable

X = the method a friend contacts you online. X is not a random variable. ( X is not numerical)

A Probability Distribution (PDF) for a Discrete Random Variable is a table, graph or formula that gives Probability of each value of X.

A Probability Distribution Function (PDF or PD) satisfies the following requirements:

1. The value X is numerical, not categorical and each P(x) is associated with the corresponding probability.

2  ΣP(X) = 1 .  A ΣP(X) = 0.999 or 1.01 is acceptable as a result of rounding.

3.  0  ≤P(x) ≤ 1 for all P(x) in the PDF.

Ex1:  PDF for number of heads in a two-coin toss are given as a table and a graph.

Table                  graph  Both are valid PDF  because Σ P(X) = 1

and each value of P(X) is between 0 and 1.

Ex2:  The number of medical tests a patient receives after entering a hospital is given by the PDF below a) Is the table a valid PDF?

The table is not a probability distribution because  Σ P(x) = 0.02+0.18+0.3+0.4 = 0.9  is not 1

b) Define the random variable x.

x = no. of medical tests a patient receives after entering a hospital.

c) Explain why the x = 0 is not in the PDF?

A patient always receives at least one medical test in the hospital.

### Parameters of a Probability distribution:

Mean μ for a probability distribution:

$$E(x) = \mu = \sum{x\cdot P(x)}$$

Variance σ2 for a probability distribution:

$$\sigma^2 = \sum{(x-\mu)^2}\cdot P(x)$$

Standard deviation for a probability distribution:

$$\sigma = \sqrt{\sum{(x-\mu)^2}\cdot P(x)}$$

To calculate Mean, variance and standard deviation of a probability distribution by technology:

Use Libretext statistics calculator:

Enter Number of outcomes, each X and P(X), calculate.

round off rule: one more decimal place than for E(x)

Two decimal places for σ and σ2.

Expected value = the long-term outcome of average of x when the procedure is repeated infinitely many times. Round to one decimal place.

Non-significant values of X.

1. The range of X from  $$\mu - 2\cdot\sigma \text{ to }\mu +2\cdot\sigma$$ are non-significant. (Range of rule of Thumb)

2. X that are outside of  $$\mu - 2\cdot\sigma \text{ to } \mu +2\cdot\sigma$$    are significant that is unlikely to occur.

Ex1: X = no. of year a new hire will stay with the company. P(x) = Prob. that a new hire with stay for x year. a) Find mean, variance, st. deviation and determine the Expected number of years a new hire will stay.

Use easycalculation.com statistics discrete random variable calculator,

Enter number of outcomes = 7.    Mean = 2.4, σ2 = 2.73, σ =1.65

The Expected no. of year a new hire will stay is 2.4 years.

b) Find probability that a new hire will stay for 4 years or more.

Add P(4), P(5) and P(6) = P( 4 or more) = 0.1 + 0.1 + 0.05 = 0.25

c) Find probability that a new hire will stay for between 3 or 5 years inclusive.

P( 3 to 5 inclusive) = 0.15 + 0.1+0.1  = 0.35

d) Find the probability that a new hire will stay for 2 years or fewer.

P(2 or fewer) = 0.12 + 0.18 + 0.30 = 0.6

e) Find the range of non-significant year of stay.

2.4 -2(1.65) to 2.4 + 2(1.65) is -0.9 to 5.7

Ex2: Given x = of number of textbooks a student buy per semester. What is the expected number of textbooks? a) Find mean, variance and standard deviation.

Use easycalculation.com statistics discrete random variable calculator, Enter number of outcomes = 6

E(x) = μ = 3.5,    σ2 = 0.61,  σ = 0.78

Expected number of textbook is 3.5 books.

b) Find Probability that a student buys at least 5 textbook.

P( at least 5)  = P(5 or more) = 0.03 + 0.02  = 0.05,

c) Find probability that x is at most 2.

P(at most 2 ) = P( 2 or fewer) = 0.02 + 0.03 = 0.05

c) Find the range non-significant.

Range of non-significant is 3.5 – 2(0.78) to 2.5 + 2(0.78) is 1.94 to 5.06.

Ch 4.1 Discrete Random Variable is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.