1. Given the random process Y(t) = At +3, (-oo
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.1. Let (X) be a simple random walk that starts from Xo = 0 and on each step goes up one with %3D probability p and down one with probability q = 1 – p. Calculate: (a) P(X, = 0), (b) EX., (c) Var(X6), (d) E(X10 | X4 = 4). (e) P(X19 = 0 | Xg = 2), (f) P(X, = 2| X 10 = 6). Consider the case p = 0.6, so q = 0.4. (g) What are EX100 and Var(X100)? (h) Using a normal approximation, estimate P(16 S X100 S 26). You should use an appropriate "continuity correction", and explain why you chose it. (Bear in mind the possible values X00 can take.)b) Let X₁, X₂, ..., Xn and Y₁, Y₂, ..., Ym be random samples from populations with moment generating functions Mx,(t) = ³t+t² and My (t) = (₁2)²5, respectively. ii) What is the value of the sample size n, if P[1(X; - X)² > 68.3392] = 0.025?
- 3. Let the random variable X have the moment generating function M(t) = What are the mean and the variance of X, respectively? -1 < t < 1.1. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).67. Let X be a discrete random variable with pmf (a) Find the pmf for Y = X². (b) Find the pmf for U = X + 2. f(x) = px(x) = 1/8, 1/4, 1/4, 1/4, 1/8, 0, x = -2 x = -1 x = 0 x = 1 x = 2 otherwise
- Let X1, X2, ..., X, be independent random variables and Y = min{X1, X2, ..., Xm}. Fy (y) = 1 – || (1 – Fx,(y)) i=1 (a) A certain electronic device uses 5 batteries, with each battery to have a life that is exponentially distributed with mean of 48 hours and is independent of the life of other batteries. If the device fails as soon as at least one of its batteries fail, what is the expected life of the device?if x be a random variable with moment generating function m,(t) = (0.6+ 0.4e*)10 then E(x)=(6) Let X be a Poisson random variable with parameter 9, then E(X²) = 1.
- (16) The moment-generating function of the geometric random variable X with parameter p is M(t) = 1-per. Use this to find the mean and variance of X.4. Let X be a random variable taking positive values and assume that E(X) exists. Which of the following statements are true? • (i) E(1+ Xª) >1+(E(X))'. (ii) E(|X|) > |E(X)|. • (iii) E(e13x) > e13E(X).Q3/(a) Let X = b(n, ) and E(x)=E(16–7x) find n. (b)Let X be a random variable with p.d.f, f(x)=c x =1,2,3,.., zero elsewhere. Find the constant c.