strate this test YOUR opportunity to deulon your mastery and understanding of th

Question

strate this test YOUR opportunity to deulon your mastery and understanding of the ma terial of this class. Work as many problems as you can, but quality matters most. Justify all answers. 20 1. a. State precise technical definitions of three of the following: random variable. (probability) distribution Px of a random variable x memory less, and probability density function. b. Use the definitions of Ex] and var (x) to show that for every discrete Rv x, var(x) Elx (ER) 2. random with parameter p e (0,1). Given its probability mass function Suppose X is a geometric variable i, calculate and a formula its probability generating function 4 x (s) Esx simplify for p(n) p (1 p)" and use this to calculate the expectation Elx (1) and variance Varlx Suppose the number of times that a person contracts a cold in any year is a Poi random variable (wi probability mass function p(n) e An /n!) with parameter A 5. Suppose a drug reduces the Pois parameter to u the population, but has no effect on the other 20%. If a person tries the d 3 for 80% of for a year and has two colds, how likely is it that the drug is beneficial for her A die is rolled twice. Find the probability mas functions (tabulate the values), calculate the expected v and sketch the graphs of the cumulative distribution functions of the random variables: X: maximum value to appear in the two rolls Y: number of prime numbers rolled. (Recall that 1 is not prime

Explanation / Answer

1>a)random variable:consider a probability space(S,B,P),where S is the sample space B is the sigma field and P is the prob function then a random variable is a real valued function X(w) deined on s and such that for every real no a, the event {w:X(w)<=a} belongs to B

Distribution Function:let X be a random varible the function F defined for all real x by

F(x)=p[X<=x]=p[w:X(w)<=x] -inf<x<inf is called the distribution function of X

Memoryless property:A random variable X is said to have the loss of memory property if the condional prob of X grater than a+b given X is grater than a is same as the unconditional probability of X grater than a+b

i.e. p[X>a+b|X>a]=p[x>=b]

Probability density function:p.d.f of the rv X is defined as f(x)=limdx--->0p[x<X<x+dx]/dx

question no 6 is not there

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