MHT-CET · Chapter

Probability Distribution

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when outcomes are equally likely;

0 \le P(A) \le 1

Locked

Probability by countingCount

n(A)

and

n(S)

with

^nC_r

: cards, balls, committeesLocked

Complement and odds

P(A') = 1 - P(A)

; odds in favour

a : b

means

P(A) = \frac{a}{a + b}

Locked

The addition theorem $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ ; just add only for mutually exclusive eventsLocked
Addition for three eventsAdd the three, subtract the pairs, add back the tripleLocked
Conditional probability $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ :
The multiplication theorem $P(A \cap B) = P(A)\,P(B \mid A)$ : draws without replacement change the second probabilityLocked
Independent events $P(A \cap B) = P(A)\,P(B)$ ; "at least one" $= 1 - P(\text{none})$
Total probability and Bayes' theorem $P(E_i \mid A) = \frac{P(E_i)P(A \mid E_i)}{\sum_j P(E_j)P(A \mid E_j)}$ :

Random variablesA random variable gives a number to each outcome of an experiment; it is discrete when its values can be listed, continuous when they fill an intervalStart
Probability mass functionA p.m.f. lists $P(X = x)$ for each value: every probability is between $0$ and $1$ and they add to $1$ ; use the total to find an unknown $k$ Locked
Writing a probability distributionList every value the random variable can take, find the probability of each from the experiment, and check that they add to 1Locked
Expected value and variance $E(X) = \sum x\,P(x)$ ; $\text{Var}(X) = \sum x^2P(x) - [E(X)]^2$ ;

Probability density functionA p.d.f. has $f(x) \ge 0$ and $\int f(x)\,dx = 1$ over its range; $P(a < X < b) = \int_a^b f(x)\,dx$ , the area under the curveLocked
Cumulative distribution function $F(x) = P(X \le x)$ : a running total of the p.m.f. for a discrete variable, $\int_{-\infty}^{x} f(t)\,dt$
Expected value of a continuous random variable $E(X) = \int x\,f(x)\,dx$ and $\text{Var}(X) = \int x^2 f(x)\,dx - [E(X)]^2$ ,

B

has happened, so

B

is the new sample space

Locked

work backwards from the result

Locked

the standard deviation is

\sqrt{\text{Var}(X)}

Locked

for a continuous one

Locked

with integrals in place of the sums

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