Ch 4.1 Discrete Random Variable
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.