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).

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

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.

Find

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

cdf definition

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

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 \(V = Vaccinated\)

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

Joind PF Properties

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

Coin-Die

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

• 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