Given that X is a random variable having a Poisson(X) distribution, compute the following: (a) P(X= 1) when λ = 3.5 P(X = 1) = (b) P(X ≤ 9) when A= 5 P(X ≤ 9) = (c) P(X> 3) when λ = 4 P(X > 3) = (d) P(X < 1) when X = 5 P(X < 1) =
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- A product is classified according to the number of defects, it contains (X1) and the factory that produces it (X2),. The joint probability distribution is given by X/X2 2 0 1/8 1/16 I/16 1/16 3/16 1/8 3 1/8 1/4 (a) Find the marginal distribution of X1Suppose X is a discrete random variable which only takes on positive integer values. For the cumulative distribution function associated to X the following values are known: F(18)=0.4F(25)=0.44F(33)=0.5F(41)=0.53F(46)=0.56F(52)=kF(57)=0.64F(65)=0.67 Assuming that Pr[25<X≤52]=0.17, determine the value of k.Let Y be a Poisson random variable with mean λ = 2. (a) Find P(Y ≥ 2). (b) Find P(Y ≥ 4|Y ≥ 2).
- Assume that X is a random variable whose conditional distribution given the variable Y is poisson P (X | Y) = Po (Y). Suppose further that Y has a gamma distribution Y ∼ Gamma (1, 1). (a) Determine the value E (XY). (b) Determine the conditional distribution P (Y | X).Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)For a random variable, its hazard function also referred to as the instantaneous failure rate is defined as the instantaneous risk (conditional probabilty) that an event of interest will happen in a narrow span of time duration. For a discrete random variable X, its hazard function is defined by the formula hX(k) =P(X=k+ 1|X > k) =pX(k)1−FX(k). For a Poisson distribution with λ= 4.2, find hX(k) and use R to plot the hazard function.
- (50) Let X be a random variable with p.d.f. k 1A random variable X is distributed as Poisson distribution with λ = 3.5. Usethis Poisson distribution (Table A.3) to determine the following probabilities: (a)P(X < 5), (b) P(2 ≤ X ≤ 6), (c) P(X > 7), (d) P(X ≥ 5).Suppose that the random variableX has the continuous uniform distribution ( 1,0 < x < 1 0, otherwise Suppose that a random sample of n = 15 observations is selected from this distribution. What is the approximate probability distribution of X-4? Find the mean and variance of this quantity. Note: Answer can be input as a fraction. f(x) = V distribution with mean = and variance =The maintenance department in a factory claims that the number of breakdowns of a particular machine follows a Poisson distribution with a mean of 2 breakdowns every 428 hours. Let x denote the time (in hours) between successive breakdowns. (a) Find λ and Ux. (Write the fraction in reduced form.) ux = f(x) = 214 (b) Write the formula for the exponential probability curve of x. P(x <4) ✔ Answer is complete and correct. 1 P(115Suppose X is a discrete random variable which only takes on positive integer values. For the cumulative distribution function associated to X the following values are known: F(23) 0.34 F(29) = =0.38 F(34) 0.42 F(39) 0.47 F(44) = 0.52 F(49) 0.55 F(56) = 0.61 = Determine Pr[29Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON