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CHAPTER THREE: UNIVARIATE DISTRIBUTIONS 1.1. DISCRETE DISTRIBUTIONS In this chapter we shall study some of the probability distributions that figure most prominently in statistical theory and applications. We shall also study their parameters, that is, the quantities that are constants for particular distributions but can take on different values for different members of families of distributions of the same kind. The most common parameters are the lower moments, mainly and . 1.1.1. ...

MOMENT GENERATING FUNCTION (m.g.f) The moment generating function (m.g.f) of a random variable X is defined as: is the expected value of . We are required to show that: and Proof We know that is differentiable. when t = 0; when t = 0; But we know that;   Example 2.8 Consider the random variable X whose p.d.f. is where is the parameter of the distribution. (a) Obtain the moment generating function of X. (b) Hence obtain the mean and variance of X...

Properties of Expectation Let X be a random variable and c a constant. Then we have: (i) E (c) = c, for c a constant (ii) E [c g (x)] = c E [g (x)], where c is a constant (iii) E [c1 g1 (x) + c2 g2 (x)] = c1 E [g1 (x)] + c2 E [g2 (x)] where c1 and c2 are constants. Proof We shall assume that the random variable X is continuous with probability density function . (i) since is a p.d.f. (ii) . (iii) By definition; If the random variable X is discrete, replace integral notati...

As in the discrete case, there are many problems in which it is of interest to know the probability that the value of a continuous random variable X is less than or equal to some real number x. Thus, let us make the following definition analogous to Definition 1.3. Definition 1.6 (Distribution Function or Cumulative Density Function) If X is a continuous random variable and the value of its probability density function at t is , the function given by for is called the Distribution Fu...

CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...

Kaplan GradReady: How to Fund your Education Assessment Question and Answer 2023. A(n) ____________ letter outlines what types of financial aid you can receive. Correct Answer: Award Complete the sentence. PLUS, Loans... Correct Answer: Are not need-based so even borrowers with higher incomes are eligible. TRUE or FALSE: A Subsidized Direct Loan requires a credit check, but an Unsubsidized Direct Loan does not. Correct Answer: False What information is needed in order to complete the F...

Knopman Marks Securities Industry Essentials Flash Cards (SIE) Question and Answer 2023. Do the following Investment Companies/Products offer continuous share offering or fixed share counts? Mutual Funds Closed-End Funds UIT ETF Correct Answer: Mutual Funds: Continuous Closed-End Funds: Fixed UIT: Fixed ETF: Fixed What Term signifies that a share

Knopman Marks Securities Industry Essentials Flash Cards (SIE) Question and Answer 2023. Do the following Investment Companies/Products offer continuous share offering or fixed share counts? Mutual Funds Closed-End Funds UIT ETF Correct Answer: Mutual Funds: Continuous Closed-End Funds: Fixed UIT: Fixed ETF: Fixed What Term signifies that a share of common stock can be freely transferred, assigned, or delivered to another entity? a) Fungible b) Negotiable c)Portable d)Endorsable...

Knopman Exam Question and Answer 2023. Registration as an investment adviser representative is always Correct Answer: at the state level For an Administrator to deny or revoke a registration it must have 2 reasons to do so, with one of the reasons being that it is in the public's best interests. Reasons include insolvency (meaning the firm cannot meet its financial obligations), conviction of a securities or money related crime within the last 10 years, or if another state prohibits the fir...

Knopman Marks Question and Answer 2023. Trust Indenture Correct Answer: Covenants, Cross Default Insider Trading / Securities Fraud Enforcement 1988 Correct Answer: Broker-dealers, 2 prong defense, rules in place, willfully ignored, unsolicited exception Regulation FD Correct Answer: On purpose: simultaneous, accident: 24 hours Fraud and Market Manipulation (Market) Correct Answer: Falsely claim market exists Statutory Disqualification Grounds Correct Answer: Felony of any kind, sec...