Example 5.10. (a) A random variahle X is distributed at random between the valuesuna I so that its probability density function is : f(x) = kx² (1 – x³), where k is a constant. Finathe value of k. Using this value of k, find its mean and variance .(b) A variable X is distributed at random between the values 0 and 4 and its probabilitydensity function is given by: f(x) = kx³ (4 – x)².%3DFind the value of k, the mean and standard deviation of the distribution.

Answered: Image /qna-images/answer/eaf68908-311a-44ee-8285-1e7bb9f199ee.jpg

Answered: Example 5-10. (a) A random variable X…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.6: Exponential And Logarithmic Equations

Problem 64E

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Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such that the initial population at time t=0 is P(0)=P0. Show algebraically that cP(t)P(t)=cP0P0ebt .
Question 1 : Suppose that the probability density function (p.d.f.) of the life (in weeks) of a certain part is f(x) = 3 x 2 (400)3 , 0 ≤ x < 400. (a) Compute the probability the a certain part will fail in less than 200 weeks. (b) Compute the mean lifetime of a part and the standard deviation of the lifetime of a part. (c) To decrease the probability in part (a), four independent parts are placed in parallel. So all must fail, if the system fails. Let Y = max{X1, X2, X3, X4} denote the lifetime of such a system, where Xi denotes the lifetime of the ith component. Show that fY (y) = 12 y 11 (400)12 , y > 0. Hint : First construct FY (y) = P(Y ≤ y), by noticing that {Y ≤ y} = {X1 ≤ y} ∩ {X2 ≤ y} ∩ {X3 ≤ y} ∩ {X4 ≤ y}. (d) Determine P(Y ≤ 200) and compare it to the answer in part (a)
2 Given the exponential distribution: f(x) = {1/θ e^(-x/θ) ; x > 0 0; ew]
2.5.8 The random variable X measures the concentration ofethanol in a chemical solution, and the random variable Ymeasures the acidity of the solution. They have a jointprobability density function f(x, y) = A(20 - x - 2y)
For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
If X is a continuous variable in the range 3 > X > 0 and its distribution function is as follows: F ( x ) = k : ( x3 + x2) find the probability density function?
1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.
suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)
1. A random variable Y has a probability density function f(y) = A(y − 1.5) over the state space −2 ≤ y ≤ 3, where A is a constant. Calculate the median of the random variable Y. 2. Suppose the lifetime of a particular type of old mobile, X, can be modelled by the probability density function f(x) = 0.5 e −0.5 x; x ≥ 0. Suppose that you just bought 5 old mobiles, what is the probability that at least 3 of the mobiles will survive less than 2 years?
If the probability mass function of the variable X described in the table is. find variance Y=x^2+4x If you know that the torque is of the second order of the variable X about the origin is equal to 2.85
QUESTION 10 Suppose f(x) = 1/4 over the range a ≤ x ≤ b, and suppose P(X > 4) = 1/2. What are the values for a and b? a. 2 and 6 b. Cannot answer with the information given. c. 0 and 4 d. Can be any range of x values whose length (b − a) equals 4. QUESTION 11 The probability density function, f(x), for any continuous random variable X, represents: a. all possible values that X will assume within some interval a ≤ x ≤ b. b. the probability that X takes on a specific value x. c. the height of the density function at x. d. None of these choices. QUESTION 12 Which of the following is true about f(x) when X has a uniform distribution over the interval [a, b]? a. The values of f(x) are different for various values of the random variable X. b. f(x) equals one for each possible value of X. c. f(x) equals one divided by the length of the interval from a to b.…
2.2 The demand for a product varies from month to month. Based on data from past years, the following probability density function shows the probabilities of MNM company’s monthly demand. Probabilities of MNM company's monthly demand Unit Demand P(X=x) 1200 0.19 2100 0.30 3300 0.40 3800 0.11 c) Calculate the standard deviation. d) Each unit produced costs the company $8.00, and each unit is sold for $25.00. How much will the company gain or lose in a month if they stock the expected number of units demanded but sell 2100 units?

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