Stat 110 - Cheat Sheet printable pdf download

STAT 110 – Cheat Sheet

Let X be a discrete r.v. then,

TOPICS

E[X] =

xP (X = x)

Counting:

Linearity: Let X, Y be r.v. and let a be a ﬁxed constant

Multiplication rule: n

. . . n

outcomes if experiment

then,

has r steps and i

step has n

outcomes.

E[aX + Y ] = aE[X] + E[Y ]

Ordered, without replacement: n(n

1)(n

2)...(n

k +

1) =

ways to choose k items out of n without

Let X be a discrete r.v. then,

(n k)!

replacement, if the order in which the k items are chosen

V ar(X) = E[(X

E[X])

] = E[X

]

E[X]

matters (e.g., out of 10 people, choose a president, vice-

president, and treasurer)

Variance is NOT linear, X, Y be r.v. and let a be a ﬁxed

Unordered, without replacement:

ways to

(n k)!k!

constant then,

choose k items out of n without replacement if order does

not matter (e.g., out of 10 people, choose a team of 3).

V ar(X + Y ) = V ar(X) + V ar(Y ) (Unless X and Y are indep)

Ordered, with replacement: n

ways to choose k items

V ar(aX) = a

V ar(X)

out of n with replacement if order matters (e.g., roll 8

dice and place them in a line: n = 6, k = 8)

Bernoulli Distribution

n+k 1

Unordered, with replacement:

ways to choose k

Bern(p) and has PMF

items out of n with replacement if order does not matter

(e.g., shake 8 dice in a cup and pour them onto the table)

(X = 1) = p and P (X = 0) = 1

Probability

Let X be the outcome of one experiment that results in

Naive deﬁnition:

either success (with probability p) or failure (with prob-

ability 1

p), then X

Bern(p).

number of outcome favorable to A

P (A) =

number of outcomes

Mean: E[X] = p

Assumes ﬁnite sample space, all outcomes equally likely.

Binomial Distribution

Axiomatic deﬁnition: P is a function from the sample

Bin(n, p) and has PMF

space S to the interval [0, 1], such that P ( ), P (S) = 1

and for disjoint A P (

) =

P (A

)

n k

P (X = k) =

for k

0, 1, . . . , n

Conditional Probability

P (A

Let X be the number of successes out of n independent

P (A B) =

P (B)

Bernoulli experiments, each with probability p of success,

then X

Bin(n, p).

and in general P (A B) = P (B A).

Mean: E[X] = np

Bayes rule

Geometric Distribution

P (B A)P (A)

P (A B) =

P (B)

Geom(p) and has PMF

P (B A)P (A)

P (B A)P (A) + P (B ¯ A)P ( ¯ A)

P (X = k) = p(1

for k

0, 1, 2, . . .

Random Variables (r.v.)

Let X be the number of failures before the ﬁrst success

in a sequence of Bernoulli experiments, each with prob-

A random variable is a function that takes every outcome

ability p of success, then X

Geom(p).

in the sample space and assigns to it a real number. It

is a numerical summary of the experiment.

1 p

Mean: E[X] =

The distribution of a r.v. speciﬁes the probability that

Hyper-Geometric Distribution

the r.v. will take on any given value or range of values.

For a discrete r.v., we can know the distribution by know-

HGeom(w, b, n) and has PMF

ing the probability mass function (PMF), P (X = x) for

all x.

n k

P (X = k) =

for k

0, 1, 2, . . .

w+b

Expectation and Variance

Stat 110 - Cheat Sheet

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