| Abdullah Al Mahmud | www.statmania.info |
Guess which
\(f(x) \to I \to D \to f(x)\)
Test with \(f(x) = x^2\)
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 | \(\frac 1 7\) | \(\frac 2 7\) | \(\frac 1 2\) | \(\frac 1 7\) | \(\frac 1 7\) | \(\frac 1 7\) |
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
Given, \(P(x) = \frac{2x+k}{56}; x = -3, -2, -1, 0, 1, 2, 3\)
Discrete or Continuous?
Clue: If \(S=\{H,T\}, P(H) + P(T) = 1\),
i.e. \(\sum\) P(All possible values) = 1
Probability Density Function (continuous)
Probability Mass Function (discrete)
\(f\left( x \right) = \frac{{{x^3}}}{{5000}}\left( {10 - x} \right)\) \(for 0 \le x \le 10\) and \(f\left( x \right) = 0\) for all other values.
\(f(x) = k(x+1); 0\lt x \lt 1\)
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
\(F_X(x) = P(X\le x)\)
Discrete
\[F(x) = \sum_{X\le x} P(x)\]
Continuous
\[\begin{eqnarray} F(x) = \begin{cases} x^2/2, & 0\le x \le 1 \\ 0, & \text{otherwise} \end{cases} \end{eqnarray}\]
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
Let, \(I = Infected\), and \(V = Vaccinated\)
\(I\) | \(\bar I\) | Total | |
---|---|---|---|
\(V\) | 3 | 276 | 279 |
\(\bar V\) | 66 | 473 | 539 |
Total | 69 | 749 | 818 |
Answers
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.
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 |
Consider the previous table
Joint probability: \(P(x_i, y_j); i = 1,2, \cdots m; j = 1,2, \cdots n\)
Marginal probability \(\to P(x_i) \leftarrow P(x_i, y_j)\)
Summing marginal probabilities will give 1.
\(P(x,y) = \frac{x+y}{9}; x=0,1,2; y = 0, 1\)
Like Bayes Theorem
\(P(X_i|y_j) = \frac{P(x_i,y_j)}{P(y_j)}; P(y_j) \gt 0\)
Properties
\(P(x,y) = \frac{x+y}{9}; x=0,1,2; y = 0, 1\)
Find \(P(X|Y)\) and \(P(Y|X)\)
Find for continuous X as well.
\(f(x) = kx^2+kx+\frac 1 8; 0 \lt x \lt 2\)