Question

In: Math

Suppose the probability mass function of a random variable X is given by ??x−1?pr(1−p)x−r, ifx=r,r+1,r+2,... f(x)...

Suppose the probability mass function of a random variable X is given by

??x−1?pr(1−p)x−r, ifx=r,r+1,r+2,... f(x) = r−1

0, otherwise

If this is the case then we say X is distributed as a Negative Binomial Random Variable with parameters r and p and we write X ∼ NegBin(r, p) (a) If we set r = 1, what distribution do we get? (b) Explain what this random variable models and justify the formula. (Hint: See Section 4.8.2 in Ross.) Math 241 Quiz 3 - Page 2 of 2 August 2019 (c) For this random variable, what is E[X] and Var[X]? There is no need to prove your answer or show any work for this part. (Hint: See Section 4.8.2 in Ross.) (d) In tossing a fair die repeatedly (and independently on successive tosses), find the proba- bility of getting the third “1” on the xth toss. (Hint: Let X denote the number of tosses required until we get our third “1”, or equivalently our third success. Then X is what kind of random variable?) (e) In tossing a fair die repeatedly (and independently on successive tosses), find the proba- bility of getting the third “1” on the fifth toss. (f) What is the average number of trials it will take to get our third “1”? (Hint: Use the results of part (c) for your solution).

Expert Solution

The PMF of Negative Binomial random variable is

This is the probability of getting the r-the success in the x-the toss.

a) When , the PMF is

$f(x)=\binom{x-1}{1-1}p^1(1-p)^{x-1}\\ f(x)=p(1-p)^{x-1};x=1,2,3,...$

This is a Geometric distribution with parameter .

b)This is the probability of getting the the first success in the x-the toss. The success probability in any toss is independent of any other tosses.

c) The expected value and variance of the Negative Binomial random variable are

d) Here . Then,

X is a Negative Binomial random variable.

e) Here , then

f) We need to find the expected value of the Negative Binomial distribution.

milcah answered 1 year ago

1. The probability mass function of a discrete random variable X is defined as p(x) =...

1. The probability mass function of a discrete random variable X is defined as p(x) = ax for x = 1, 2, 4, 8 (p(x) =0 for all other values) then the value of a is? 2. Let X be a discrete random variable with Var(X) =6.0 and E(X2) = 17.00. Then: E(X) = ? 3. If X is a binomial random variable with parameters n and p, i.e. X ~ b(x; n, p), then the expected value of X...

A discrete random variable named X has the following pmf (probability mass function): X P(x) 1...

A discrete random variable named X has the following pmf (probability mass function): X P(x) 1 0.6 3 0.2 7 0.1 11 0.1 17a) Find P(X>6) 17b) What is the probability two independent observations of X will both equal 1? 17c) Find the population mean also known as E(X)= the expected value of 17d) Find the population variance of X with both the “regular” formula and “convenient” formula Make a table and show your work. You should get the same...

A probability distribution function P(x) for a random variable X is defined by P(x) = P...

A probability distribution function P(x) for a random variable X is defined by P(x) = P r{X ≤ x}. Suppose that we draw a list of n random variables X1, X2, X3 · · · Xn from a continuous probability distribution function P that is computable in O(1) time. Give an algorithm that sorts these numbers in linear average case time.

2 Consider the probability density function (p.d.f) of a continuous random variable X: f(x) = (...

2 Consider the probability density function (p.d.f) of a continuous random variable X: f(x) = ( k x3 , 0 < x < 1, 0, elsewhere, where k is a constant. (a) Find k. (b) Compute the cumulative distribution function F(x) of X. (c) Evaluate P(0.1 < X < 0.8). (d) Compute µX = E(X) and σX.

Question 1: Given the following probability distribution for a random variable X: x P(X=x) -2 0.30...

Question 1: Given the following probability distribution for a random variable X: x P(X=x) -2 0.30 -1 0.15 0 0.20 1 0.20 2 0.15 a) Explain two reasons why the above distribution is a valid probability distribution. b) Calculate μX and σX. c) Determine the cdf(X), and write it as an additional column in the table. d) Calculate P(−1<X≤3) . e) Draw a histogram that represents the probability distribution of X.

Suppose R is a binomial random variable with r=2 trials and success probability p=0.5. Let A...

Suppose R is a binomial random variable with r=2 trials and success probability p=0.5. Let A and B be uniform random variables on [0,1] and that A, B, R are mutually independent. Find PDF for Z=RAB. (Solve P(Z<=z) for z ∈[0,1] by conditioning on the value of R ∈{0,1,2}.)

Given a binomial random variable with n = 100 and p = .5, estimate the Pr[X...

Given a binomial random variable with n = 100 and p = .5, estimate the Pr[X ≥ 40]

Consider a discrete random variable with the following probability mass function x 0 1 2 3...

Consider a discrete random variable with the following probability mass function x 0 1 2 3 4 5 p(x) 0.1 0.1 0.2 0.3 0.2 0.1 Generate a random sample of size n =10000 from this distribution. Construct a bar diagram of the observed frequencies versus the expected frequencies.(using R)

Suppose X and Y are {0, 1}-valued random variables with joint probability mass function given in...

Suppose X and Y are {0, 1}-valued random variables with joint probability mass function given in the table below p(x, y) y 0 1 x 0 0.3 0.2 1 0.1 0.4 a. Determine E(Y |X = x) and var(Y |X = x). b. Use part a) to find E(Y ), and compare with the result using the marginal distribution of Y . c. Use part a) to find var(Y ), and compare with the result using the marginal distribution of...

Consider a continuous random variable X with the probability density function f X ( x )...

Consider a continuous random variable X with the probability density function f X ( x ) = x/C , 3 ≤ x ≤ 7, zero elsewhere. Consider Y = g( X ) = 100/(x^2+1). Use cdf approach to find the cdf of Y, FY(y). Hint: F Y ( y ) = P( Y <y ) = P( g( X ) <y ) =