Probability (Random Variable)

Abdullah Al Mahmud

Previous Chapter

Introduction to Probability

Concepts

Random Variable

  • Discrete
  • Continuous

Guess which

  • No. of people dying each day
  • Number of heads in successive tosses.
  • Heights of people in Bangladesh
  • GPA of students
  • Grade of students in individual subjects
  • Income tax paid by people

Integration

  • \(\int x^3+2x\)
  • \(\int_2^3 x^2+x\)
  • Relationship between integration (I) and differentiation(D).
Answer

Probability Distribution

Example

Results of an unbiased die throw

x 1 2 3 4 5 6
P \(\frac 1 6\) \(\frac 1 6\) \(\frac 1 6\) \(\frac 1 6\) \(\frac 1 6\) \(\frac 1 6\)

Biased

x 1 2 3 4 5 6
P \(\frac 1 7\) \(\frac 2 7\) \(\frac 1 2\) \(\frac 1 7\) \(\frac 1 7\) \(\frac 1 7\)

Number of Heads in Coin Toss

Two coins are tossed successively.

N:B: X is variable & x is value

X = number of heads in the coin toss

x = 0, 1, 2

  • P(x=0) = ?
  • P(x=1)=?, P(x=2)=?

Probability Function

Given, \(P(x) = \frac{2x+k}{56}; x = -3, -2, -1, 0, 1, 2, 3\)

Discrete or Continuous?

  1. k = ?
  2. Find probability of each value of x
  3. Find \(P(-2 \le x \le 2)\)

Clue: If \(S=\{H,T\}, P(H) + P(T) = 1\),

i.e. \(\sum\) P(All possible values) = 1

pdf and pmf

Probability Density Function (continuous)

  • \(f\left( x \right) \ge 0\) for all x
  • \(\displaystyle \int_{{\, - \infty }}^{{\,\infty }}{{f\left( x \right)\,dx}} = 1\)
  • \(P\left( {a \le X \le b} \right) = \int_{{\,a}}^{{\,b}}{{f\left( x \right)\,dx}}\)
  • \(P(X=x)=0\) (theoretically, why?)

Probability Mass Function (discrete)

  • \(P\left( x \right) \ge 0\) for all x
  • \(\displaystyle \sum_{{\, - \infty }}^{{\,\infty }}{{f\left( x \right)\,dx}} = 1\)
  • \(P_x(X) = P(X=x)\)

pdf vs pmf curves

Pdf problem 01

\(f\left( x \right) = \frac{{{x^3}}}{{5000}}\left( {10 - x} \right)\) \(for 0 \le x \le 10\) and \(f\left( x \right) = 0\) for all other values.

  1. Show that \(f\left( x \right)\) is a probability density function.
  2. Find \(P\left( {1 \le X \le 4} \right)\)
  3. Find \(P\left( {x \ge 6} \right)\)
  • is it \(\displaystyle \int_{7}^{10} f(x) dx\)?

Pdf problem 02

\(f(x) = k(x+1); 0\lt x \lt 1\)

  1. \(P(X=2)=?\)
  2. \(k=?\)
  3. \(P(0.4 \lt X \lt 2)=?\)
  • \(\int_{0.4}^1 f(x) + \int_{1}^2 f(x) \to 0\)

Cumuluative Distribution Function

F(x) or cdf accumulates all of the probability less than or equal to.

x 1 2 3 4 5 6
P (x) \(\frac 1 7\) \(\frac 1 7\) \(\frac 1 2\) \(\frac 1 7\) \(\frac 1 7\) \(\frac 1 7\)
F (x) \(\frac 1 7\) \(\frac 2 7\) \(\frac 4 7\) \(\frac 5 7\) \(\frac 6 7\) \(1\)

Find

  1. \(P(X<4)\)
  2. \(P(3<X<6)\)

cdf definition

\(F_X(x) = P(X\le x)\)

Discrete

\[F(x) = \sum_{X\le x} P(x)\]

Continuous

  • \[F_{X}(x) = \int_{-\infty}^x f_X(t)dt\]
  • Find cdf for \(f(x) = 2x; 0\le x \le 1\)
  • \(\int \to x^2\)
Answer

cdf properties

  • \(P(a\le x \le b) = F(b)-F(a)\)for \(a\lt b\); what if \(a \lt x \lt b?\)
  • For continuous x, \(f(x) = \frac{d}{dx}[F(x)]\)
  • \(F(-\infty) =0 , F(+\infty) = 1\)

Joint Probability Function

Let, \(I = Infected\), and \(V = Vaccinated\)

\(I\) \(\bar I\) Total
\(V\) 3 276 279
\(\bar V\) 66 473 539
Total 69 749 818

Find the probability that

  1. a vaccinated person is infected
  2. a non-vaccinated person is uninfected
  • These are joint probabilities \(\to\) P(x,y)
  • \(P(x,y) =P(x) \cdot P(y)\) if \(x\) and \(y\) are independent.

Joint-Marginal-Conditional

Let, \(I = Infected\), and \(V = Vaccinated\)

\(I\) \(\bar I\) Total
\(V\) 3 276 279
\(\bar V\) 66 473 539
Total 69 749 818
Find the probabilities that
  1. a vaccinated person is infected
  1. a non-vaccinated person is uninfected
  1. vaccinated if infected
  1. infected if not vaccinated
  1. vaccinated
  1. uninfected

Answers

Joind PF Properties

  • \(P(x,y) \ge 0\)
  • \(\Sigma\Sigma P(x,y)=1\)

Coin-Die

Die/Coin 1 2 3 4 5 6
H (1) H1 H2 H3 H4 H5 H6
T (0) T1 T2 T3 T4 T5 T6

X = Outcome of coin toss

Y = Outcome of die throw

x = 0, 1; y = 1, 2, 3, 4, 5, 6

Construct the distribution.

Joint-Marginal-Conditional Revisited

Exam (X) \(\to\)
Result (Y) \(\downarrow\)
PSC JSC SSC HSC Total
Passed 30 26 23 25 104
Failed 12 13 10 14 49
Absent 5 2 3 4 14
Total 47 41 36 43 167
  • Marginal: \(P(Pass) = P(x_1)=P(x_1,y_1)+P(x_1,y_2)+P(x_1,y_3)\)
  • \(P(Absent) = P(x_3) = P(x_3,y_1)+P(x_3,y_2)+P(x_3,y_3)\)

Marginal Probability

Consider the previous table

Joint probability: \(P(x_i, y_j); i = 1,2, \cdots m; j = 1,2, \cdots n\)

Marginal probability \(\to P(x_i) \leftarrow P(x_i, y_j)\)

  • For x: \(\displaystyle P(x_i) = \sum_{j=1}^n P(x_i, y_j); i = 1,2, \cdots m\)
  • For y: \(\displaystyle P(y_i) = \sum_{i=1}^m P(x_i, y_j); j = 1,2, \cdots n\)
  • What about continuous x?

Marginal Probability Properties

  • \(P(x_i) \ge 0\) and \(P(y_i) \ge 0\)
  • \[\sum_{i=1}^m P(x_i)=\sum_{j=1}^n P(y_j)=1\]

Summing marginal probabilities will give 1.

Joint-Marginal Example

\(P(x,y) = \frac{x+y}{9}; x=0,1,2; y = 0, 1\)

  • Find marginal probabilities
  • Check properties (sum)
  • \(P(x) = \frac{2x+1}{9}\)

Conditional Probability Function

Like Bayes Theorem

\(P(X_i|y_j) = \frac{P(x_i,y_j)}{P(y_j)}; P(y_j) \gt 0\)

Properties

  • \(\sum_{j=1}^m P(x_i|y_j)=\sum_{i=1}^m P(y_j|x_i)=1\)

Conditional Probability Example

\(P(x,y) = \frac{x+y}{9}; x=0,1,2; y = 0, 1\)

Find \(P(X|Y)\) and \(P(Y|X)\)

Find for continuous X as well.

Find k for pdf

\(f(x) = kx^2+kx+\frac 1 8; 0 \lt x \lt 2\)

  1. Find k
  2. Find \(P(1 \lt X \lt 2)\)

Next Chapter

Mathematical Expectation