2. a) Let X and Y be independent identically distributed binomial randomvariables, X~ Bin(3, 0.4) and Y ~ Bin(3, 0.4).i) Compute the probability P(X = 0).ii) Compute the probability that the product XY equals 0, that is,compute P(XY = 0).b) The actual amount of jam (in g) that a filling machine puts into "150 g"jars may be looked upon as a random variable having a normal distribu-tion with σ = 6 and mean μ = 155.What proportion of jars, in the long-run, contain less than 150 g?N(0, 1),Give your answer in terms of the function, where for Z(z) = P(Z), and a give a sketch of the normal curve with a shadedarea that corresponds to the required probability.

Let's solve the provided problems step by step:### 2(a)#### i) Compute the probability \( P(X = 0)…

Answered: 2. a) Let X and Y be independent…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A X is a normally distributed random variable with u = 25 and o = 5. a) What is the 85th percentile of X? b) What is P(X > 30)? c) What is P(X = 25)?
4. Let X₁ ~ N(0, 3²), X₂ ~ N(4,4²), and X3 ~ N(-1,22) be independent random variables. Find P (X₁ + X₂ + 3X3 < −8), using the table for standard normal distribution.
The probability that an event A occurs is 0.1. When A occurs, X is a discrete random variable with uniform distribution among the possible values {-1,0, 1}. When A does not occur, X is a discrete random variable with uniform distribution among the possible values {-2, –1,0, 1, 2}. What is P(X 2 0)? a) 0.563 b) 0.607 c) 0.800 d) 0.872 e) 0.918
a.b.c
An instant lottery ticket costs R3. Out of a total of 10000 tickets printed for this lottery, 1000 tickets contain a prize of R15 each, 100 tickets contain a prize of R60 each, 5 tickets have a prize of R3000 each, and 1 ticket have a prize of R15000. Let be the random variable that denotes the total amount of money a player wins by playing lottery. Assume that you bought 1 ticket. What is the probability that you will win R60?
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a) Find the value of the constant C. b) Find the variance of X.
Exercise 4. Let = {a,b,c,d} and define a probability measure on 222 by P(a) = P(c) = 1/6 and P(b) = P(d) = 1/3. Define random variables X and Y by X(a) = X(b) = 2, X(c) = X(d) = −2 Y(a) = Y(d) = -1, Y(b) = Y(c) = 1. (a) List all sets in σ(Y). (b) Let Z = E(XY), determine Z(a) Z(b), Z(c) and Z(d).
3) A random variable X takes the values 0, 2 and 3 with probabilities 0.3, 0.1 and 0.6, respectively. A random variable in the form of Y = 3(X-1)2 is defined. a) Find the expected value and variance of the random variable X. b) Find the variance of the Y random variable. c) Find the cumulative distribution function of the Y random variable.
36 - Calculate the variance of X, Var (X) =?, By finding the value a in the table so that the X random variable given the probability distribution table below is the probability distribution.A) 2.93B) 2.85C) 3.10D) 3.25E) 2.71
An environmental engineer collected 10 moss and 10 lichen specimens. The engineer instructs a laboratory intern to randomly select 15 of the specimens. The probability mass function of the number of lichen specimens selected at random is: a) H (x; 15; 10; 20) b) P (x; 10) c) Bnegative (x, 10, 0.666) d) B (x, 10, 0.666)
23. Let Y denote a geometric random variable with probability of success p. (a) Show that for a positive integer a, P(Y > a) = (1 − p)a (b) Showthatforapositiveintegeraandb,P(Y >a+b|Y >a)=(1−p)a =P(Y >b). (c) Why do you think the property in (b) is called the memoryless property of the geometric distribution? (d) In the development of the distribution of the geometric random variable, we assumed that the experiment consisted of conducting identical and independent trials until the first success was observed. In light of these assumptions, why is the result in part (b) “obvious”? (e) Show that P (Y = an odd integer) = p 1−q2 24. Given that the random variable W is binomial distribution with n trials and success probability p in each trial and P (W = w) = h(w), show that (a) h(w) =p(n−w+1),w>0 h(w − 1) (1 − p)w (b) E [W(W − 1)]= n(n − 1)p2. ? 1 ? 1−(1−p)n+1 (c)E W+1 = (n+1)p

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