MATH 3122 Stats Exam 2 Prep

What are the properties of the pdf?

a) f(x)≥0, x∈S b) ∫f(x)dx=1, integrated over the x∈S c) If (a,b) is the interval that contains the entire or a part of the the sample space S, then the probability of the event {a<X<b} is P(a<X<b)= ∫f(t)dt, integrated over all t<0 and t=x. The bounds of the integral in part c) are from -∞<t<x.

What is the expected value and variance of a geometric distribution with a discrete RV?

µ= 1/p variance is: (1-p)/p²

What is the variance of the binomial distribution with probability p?

σ^(2) = n*p*(1-p)

What is the variance of RV X?

σ^2 = E[ (X^2) ] - ( E[X] )^(2)

What is the variance of the geometric distribution?

σ^2= (1-p)/(p^2)

What is the expected value of a variable of the continuous type?

∫x *f(x)dx for all x defined in the cdf.

What is the basic pmf for the Poisson distribution?

[ λ^(x) * e^(-λ) ]÷ x!

What is the short hand for a binomial distribution with mean n and probability of success p?

B(n,p)

If X has an exponential distribution, the the probability of X= x+ y, given that it has lasted at least x units, is exactly the same probability of X=x.

P(X>b | X>a)= P(X>b) ÷P(X>a)= P(X>(b-a))

What is the expected value of a bivariate regression?

E(x+y) = ΣΣf(x,y) for all (x,y) in the space of x and y; or in general for some function of x and y, u: E(u(x,y))= ΣΣu(x,y)P(X=x, Y=y) for all (x,y) in the space of x and y.

What is the variance of a bivariate regression?

E(x+y) = ΣΣf(x,y) for all (x,y) in the space of x and y; or in general for some function of x and y, u: E(u(x,y))= ΣΣu(x,y)P(X=x, Y=y) for all (x,y) in the space of x and y.

How would you reference a standard normal distribution with a negative z?

I(z) is the standard normal distribution cdf from values of -∞ to z. Then I(-z) = 1-I(Z)

What is the cdf of a normal distribution?

I(z)∼Standard normal distribution with parameter Z. Where I = ∫f(t)dt = ∫1/(σ√(2π)) e^(-t²/2)dt integrated from -∞ to z.

What is the definition of expected value?

If f(x) is the pmf of the random varaible X of the discrete type with space S, and if the summation ∑u(x)f(x) for all x∈S exists, then the sum is called the mathematical expectation of u which is a function of x, and is denoted by E(u(X)). E(X) = ∑xf(x) for all x∈S.

What are the properties of a uniform distribution?

If the support contains [a,b] then f(x) = 1/[b-a] which this is referred to as the rectangular distribution.

What are the properties of the variance with parameter a and constant b?

If you know the variance of X, then the variance of RV Y= aX+b is a^(2) * variance of X. The addition or subtraction of a constant from X does not change the variance. Ex; Var(X-1) = Var(X). Var(aX+b)=(a^2)Var(X)

What is the pdf of the normal distribution?

If z=(x-µ)/σ then the pdf of the normal distribution is: f(X) = 1/[ σ √(2π) ] * e^(-z²/2)

What are the properties of a binomial distribution?

It has the following properties: 1) It is performed n times, where n is a constant 2) The trials are independent 3) The probability of success on each trial is constant p; the probability of failure is q=1-p. 4) The random variable X equals the number of successes in the n trails. f(x)= nCx * (p^x)*(1-p)^(n-x)

What is the definition of the joint pmf?

Let X and Y be two RV on a discrete space. Let S denote the corresponding two-dimensional space of X and Y, the two random variables of the discrete type. The probability that X=x and Y=y is denoted by f(x,y)= P(X=x,Y=y). It has the properties of: a) 0≤f(x,y)≤1 b) ΣΣ f(x,y) =1 for all (x,y)∈S of xy c) P[(X,Y)∈A]= ΣΣ f(x,y) for all (x,y)∈A, where A is a subset of the space S of xy.

What is the definition of the moment generating function?

Let X be a RV of the discrete type with pmf f(x) and space S. If there is a positive number H that E(e^(tX))= ∑e^(tX)f(x) for all x∈S exists and is finite for - h<t<h, then the function defined by M(t) = E(e^tX) is called the moment generating function (mgf) of X or of the distribution of X. Basically M(0) =E(X)=∑xf(x) for all x∈S.

What is the correlation coefficient?

The correlation coefficient equals the cov(X₁,X₂)/(σ₁*σ₂). Also E(X*Y)= µ₁*µ₂+ η*σ₁*σ₂. where η= cov(X₁,X₂)/(σ₁*σ₂)

What is covariance?

The covariance for RVs X and Y is: E( (x₁-µ₁)(x₂-µ₂) ) = ΣΣ(x₁-µ₁)(x₂-µ₂)f₁(x₁)f₂(x₂) for all x₁ and x₂ in the space of x₁ and x₂. It is also E( X₁*X₂) -µ₁*µ₂

With a continuous variable, what is the expected value and variance for the exponential distribution?

The exponential distribution has the following formula with θ= 1/λ: f(X)= [ 1÷θ ]e^(-x/θ) E(X) = θ Var(X)= θ²

What special distribution is from the Poisson distribution?

The exponential distribution. If λ=1/θ, then the random variable X has an exponential distribution if its pdf is defined by: f(x)= (1/θ)e^(-x/θ)

What is the cumulative distribution function (cdf) for a discrete RV defined by?

The function is defined for when: F(x) = P(X≤x), -∞<x<∞;

What is the gamma function?

The gamma function is Γ(1) = ∫e^(-y)dy=1 integrated from 0 to -∞. Which if you see this function, then: Γ(n)=(n-1)!

What is the definition of the Marginal Probability mass function.

The marginal pmf of X is defined by: P(X=x)= f(x) = Σf(x,y) for all y in the space, S of y The mpmf of Y is: P(Y=y)= f(y) = Σf(x,y) for all x in the space, S of x Remember that the sum is only with respect to the opposite variable; essentially holding the variable in question constant throughout the joint pmf.

What is the mean and variance of the binomial distribution?

The mean is n*p. Variance: n(p)(1-P)

What are the properties of the probability mass function (pmf) for a discrete random variable X?

The pmf f(x) of a discrete random variable X is a function that satisfies the following roperties: a) f(x)>0, where x∈S; b) ∑ f(x) =1, for all x∈S; c) P(x∈A) = ∑f(x) for all x∈A, where A⊂S; To determine the probability associated with the event A∈S, we would sum the probabilities of the x values in A.

When are variables X and Y independent in a bivariate distribution?

The random variables X and Y are independent if and only if, for every x∈Sx and y∈Sy that: P(X=x,Y=y) = P(X=x)P(Y=y) or f(x,y)=f(x)*f(y)

What parameters does the standard normal distribution have?

The standard normal dist has µ= 0, and σ²=1 or simply wrote as N(0,1)=N(µ,σ²)

What is the variance of a continuous RV?

The variance =Var(X) = E[(X-µ)²] =∫(x-µ)²dx from −∞<x<∞

What is the expected value and the variance of the Poisson distribution?

The variance and the mean are both equal to λ.

What are the properties of expected value?

They are: a) If c is a constant, then E(c)= c. b) if c is a constant and u is a function, then E[c*u(X)]=c*E[u(x)] c) If a and c are constants and b and d are functions of x then: E[ ab(x) + cd(x)]= a*E[b(x)]+ b*E[d(x)]

What is the mean and variance if the moment generating function is: M'(t)= e^(4t+16t^(2)) ?

This is the moment generating function of the normal distribution which has the form of: M'(t)= e^(µt+ σ²t²/2) So the mean and variance of the function is: µ=4, σ²=8

When is a pdf uniform?

When all the values of x have the same value for the pdf. All values have the same probability.

What conditions must hold for to have an approximate Poisson process?

With a parameter λ>0: a) The numbers of occurrences in non-overlapping subintervals are independent b) the probability of exactly one occurrence is in a sufficiently short subinterval of length h is approximately λh. c) The probability of two or more occurrences in a sufficiently short subinterval is essentially zero.

How would you find P(a<X<b) if X has a normal distribution?

You can convert this into a standard normal distribution with values of z. Remember z= [X-µ]/σ, so: P(a<X<b) = P([a-µ]/σ < [X-µ]/σ < [b-µ]/σ) Which you can look up these values in the standard normal distribution table. for z of a=[a-µ]/σ and z of b.