X -3 -2 0 1 3 f(x) 1/2 1/10 1/5 1/10 1/10 Questions: a) Compute the expectation E(x). b) Compute the variance of x. c) Find P(x < 1) and P(−2 < x ≤ 1). d) Plot the probability function (probability mass function) of X. e) Calculate and plot the cumulative distribution function F(x).
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- A random variable X has a N(0,1) distribution. Use the moment generating function of X to compute (a) the third and (b) fourth moments of X, i.e., E(X³) and E(X4).b) Probability density function of continuous random variable X, f(x) = {C(3+ x) 1 2.5) and P (X<1.5).Exercise 3. Let X be a random variable with mean and variance o². For a € R, consider the expectation E((X-a)²). a) Write E((X-a)²) in terms of a, u and o². b) For which value a is E((X - a)²) minimal? c) For the value a from part (b), what is E((X-a)²)?
- 8/ Example 7-36. Let {Xg} be mutually independent, each assuming the values 0,1.2 3 ..., a – 1 with probability Let S„ = X1 + X2 + ... + Xp. Show that the probability generating P (s) = { 6)= { 1-s a(1 – s) function of S, is : P (S„ = j) =£ (-1)**jtev (") (;="av) and hence V=01 Let g(x)=√x, for x = 4, 9, 25, be a probability mass function. 10 a) Find the cumulative probability function G, and draw its graph. b) Is it true that G(29) = 0? Why or why not? a) Define each piece for the cumulative probability function. G(x) = for (Simplify your answer.) ▾ (Simplify your answer.) ▾ (Simplify your answer.) G(x) = G(x) = G(x) = Choose the correct graph below. for for for O A. AG(x) |4 M (Simplify your answer.) ✔ b) Is it true that G(29) = 0? Why or why not? O A. It is true because g(x) is 0 for x = 29. O B. It is true because P(X≤x) for x = 29 is 0. O C. It is not true because G(x) is not defined for x = 29. O D. It is not true because P(X≤x) for x = 29 is 1. O B. $||||$ ŏ of AG(x) 144 for of T X Q ✔ C O C. AG(x) 144 HUIS → ✔ O D. fỏ• ž AG(x) 144 • T to Q ✔Let a random sequence x(n) = Bn + A, where A, B are independent Gaussian RVs with zero expected value and variance of, of, respectively. Calculate: a) The expected value of the autocorrelation of the random sequence x(n). b) The second statistical moment of the random sequence x(n).
- A simple random sample X1, …, Xn is drawn from a population, and the quantities ln X1, …, ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false: a) X1, …, Xn come from a population that is approximately lognormal. b) X1, …, Xn come from a population that is approximately normal. c) ln X1, …, ln Xn come from a population that is approximately lognormal. d) ln X1, …, ln Xn come from a population that is approximately normal.ACTIVITY 2. Consider the probability mass functions of the two investment options that are presented to a businessperson. Compute for the mean and variance of each investment. Investment A Profit P(x) x: P(x) x2 x2- P(x) (x) 10,000 10 3 5, 000 10 -3,000 10 Ex2. P(x) Investment B Profit P(x) x2 x* - P(x) x: P(x) (지) 2 20,000 10 5 16,000 10 3 -15,000 10 Compare the measures of variability for each investment. Mean Variance Standard Deviation Investment A Investment BA random walk (RW) {Sn}n>o is a sum of id r.v.s X1, X2, •• , Xn,withP(X1 = a) = p, P(X1 = b) = 1 -p= q.Find the expectation E(Snl and variance var(Sn) of Sn for any n.
- II. Let X be a discrete random variable with possible values xi, i=1,2,.., and p(xi)=P(X=xi) is called the probability distribution function of X. We have > p(x,)=. The expected value, also called expectation, average, or mean, of X is defined as u= E(X)=, For any function g(x), x=1,2,3,.., E(g(x)) =. . The variance of X: Var(X)= Since we have E(X – µ)² = E(X² – 2µX + µ) = EX² – 2µEX + µ² = EX² – (EX)² the variance of X: Var(X)=.Let X be the number of years before a particular type of machines will need replacement. Assume that X has the probability function f(1) = 0.1, f(2) = 0.2, f(3) = 0.2, f(4) = 0.2, f(5) = 0.3. Find the probability that the machine needs no replacement during the first 3 years.An epidemic process is such that an infected individual is infectious for a time period which is exponentially distributed with rate parameter 3. While infectious, the individual gets into contact with susceptible individuals according to a Poisson process with rate parameter λ. Let X be the number of susceptible individuals who get into contact with infected individual. The probability generating function of the distribution of X is Select one: a. O b. B B+λ-As Oc. e-B(1-s) -XB(1-8) βλ B+λ-As d. e