Probability & Statistics Flashcards

Question

Sum of Probabilities for a Discrete Random Variable

Answer 1

The sum of the Outputs must equal 1

Answer 2

If X is a discrete random variable which takes values x1, x2, x3, . . ., then the expectation of X (or the expected value of X) is defined by E(X) = x1P(X = x1) + x2P(X = x2) + x3P(X = x3) + · · · .

Answer 3

m ≤ E(X) ≤ M

Answer 4

E( f(X) ) = f(x1)P(X = x1) + f(x2)P(X = x2) + f(x3)P(X = x3) + · · ·

Answer 5

The nth moment of the random variable X is the expectation E(X^n)

Answer 6

Var(X) = [x1 − E(X)]2P(X = x1) + [x2 − E(X)]2P(X = x2) + [x3 − E(X)]2P(X = x3) + ...

Answer 7

Var(X) = E(X^2) − [E(X)]^2

Answer 8

E(aX + b) = aE(X) + b

Answer 9

Var(aX + b) = a^2Var(X)

Answer 10

It is where a random variable X only takes values 0 and 1

Answer 11

E(X) = p, Var(X) = p(1 − p)

Answer 12

A discrete random variable X has the Binomial(n, p) distribution, denoted X ∼ Bin(n, p), if its p.m.f. is : P(x =k) = nCk x p^k x (1-p)^n-k

Answer 13

E(X) = np, Var(X) = np(1 − p)

Answer 14

A discrete random variable X has the Geometric(p) distribution, denoted X ∼ Geom(p), if its p.m.f. is : P(X = k) = p(1 − p)^k−1

Answer 15

E(X) = 1/p Var(X) = 1 − p/ p^2

Answer 16

the hypergeometric distribution describes the probability of successes in draws, without replacement, from a finite population P(X = k) = mCk x (n-m)C(l-k) / nCl

Answer 17

E(X) = l x (m/n) Var(X) = l x (m/n) x (n-m/n) x (n-l/n-1)

Answer 18

The negative binomial distribution models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified number of successes occurs P(X = k) = (k+r-1)C(r-1) x p^r x (1-p)^k

Answer 19

E(T) = (1-p)r/p Var(T) = (1-p)r/p^2

Answer 20

uniform distribution refers to a type of probability distribution in which all outcomes are equally likely P(X=k) = 1/(n+1) if m<=k<=n+m

Answer 21

where n = b-a, and m=a E(X) = m +(n/2) Variance = n(n+2)/12

Answer 22

Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate. P(X=k) = (λ^k/k!) x e^-λ

Answer 23

E(X) = λ, Var(X) = λ

Answer 24

The cumulative distribution function (c.d.f.) of a random variable X is the function which given t has output P(X ≤ t).

Answer 25

Let X be a discrete random variable which takes integer values. The moment generating function (mgf ) of X is the function which given t has output E(e^(tX))

Answer 26

We say that a random variable X is a continuous random variable if there exists a continuous function fx from R to [0, ∞) with the following property: P(a<= X <= b) = integral of fx(t)

Answer 27

E(X) = integral tfx(dt) Var(X) = E(X^2) − (E(X))^2

Answer 28

E(X) = (a+b)/2 Var(X) = (b-a)^2/12

Answer 29

It is often used to model the time elapsed between events P(X=t) = λe^λt

Answer 30

E(X) = 1/λ Var(X) = 1/λ^2

Answer 31

Let X and Y be two discrete random variables defined on the same sample space and taking values x1, x2, . . . and y1, y2, . . . respectively. The function: (xk, yl) → P( (X = xk) ∩ (Y = yl) )

Answer 32

P(X=xk) = suml P(X = xk, Y = yl) same goes the other way the idea is that if we only care about the probability of X taking a particular value, we need to sum over all possible values of Y

Answer 33

If g(X, Y ) is a real-valued function of the two discrete random variables X and Y then the expectation of g(X, Y ) is obtained as E( g(X, Y ) ) = sumk suml g(xk, yl)P(X = xk, Y = yl)

Answer 34

If X and Y are discrete random variables then E(X + Y ) = E(X) + E(Y )

Answer 35

Two discrete random variables X and Y are independent if the events “X = xk” and “Y = yl” are independent for all possible values xk, yl

Answer 36

The covariance of X and Y is defined by:

Answer 37

Cov(X, Y ) = E(XY ) − E(X)E(Y )

Answer 38

Using the substitution z = (x − µ)/σ one can confirm that the p.d.f. is normalised

Answer 39

When you standardise a normal distribution, the mean becomes 0 and the standard deviation becomes 1

Answer 40

Let Z ∼ N(0, 1) denote a standard normal random variable.

Answer 41

If X 1 ,X 2 ,…,X n are independent normal random variables with mean μ and variance σ2, the sum X=X1+X2+⋯+Xn is also a normal random variable. The distribution of X is given by X∼N(nμ,nσ2), meaning the mean of X is nμ and the variance of X is 2 nσ2

Answer 42

A survey is the collection of data from a sample of the population.

Answer 43

In an observational study researchers observe the behaviour of individuals without trying to influence the outcome of the study

Answer 44

In a designed experiment researchers apply some treatment to the units under investigation and measure the response.

Answer 45

Continuous variables/data are variables which are given in terms of real numbers Discrete variables/data are variables which are given by integers

Answer 46

Categorical variables/data are variables which are expressed in terms of categories Ordinal variables/data are variables which are expressed in terms of ordered categories

Answer 47

This is called the estimator of the mean where x bar is called the point estimate

Answer 48

If n is odd Q2 = x (n+1/2) If n is even Q2 = average (x(n/2),x((n/2) +1)

Answer 49

This is a biased estimator

Answer 50

mx, Q1, Q2, Q3, Mx

Answer 51

An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter

Answer 52

That means Σ2is biased and it slightly underestimates the variance

Answer 53

Measures the average squared difference between the estimated values and the actual value

Answer 54

The fraction of 1−α of such intervals contain the population mean µ. We call 1−α the level of confidence

Answer 55

A hypothesis is a statement about a parameter θ of the pmf (or pdf ) of a random variable X

Answer 56

In hypothesis testing the central claim, the null hypothesis H0 is a statement θ = θ0 which we intend to find evidence against

Answer 57

Assume the null hypothesis H0 is valid. We say a type-I error has occurred if the test procedure for H0 rejects the null hypothesis. The probability of a type-I error occurring is called the significance level α of the test procedure

Answer 58

The test procedure tests the null hypothesis H0 against the so called alternative hypothesis H1. The alternative hypothesis specifies under which conditions the null hypothesis should be rejected.

Answer 59

If we test H0 : θ = θ0 against H1 : θ /= θ0 we need a two-sided test.

Answer 60

If we test H0 : θ = θ0 against H1 : θ > θ0

Answer 61

If we test H0 : θ = θ0 against H1 : θ < θ0

Answer 62

A type-I error occurs if the test rejects the null hypothesis H0 even though H0 is valid

Answer 63

the test does not reject H0 even though in some sense H0 is false

Answer 64

The probability for a type-II error is denoted by β. 1−β is called the power of the test

Answer 65

the P-value of the observation is the probability to observe a sample statistic as extreme or more extreme as the observation under the assumption that the null hypothesis is true At significance level α the null hypothesis is rejected if the P-value obeys P < α.