Q6.(a)The continuous random variable X has a probability density function given by211

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Answered: Q6. (a) The continuous random variable…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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4. Let x be a continuous random variable with the following probability density function 3 (2 – x) 0 < x< 2 f (x) = else (a) Find E(x) (b) Find Var(x)
1. A continuous random variable X has a probability density function (PDF) p(x) = k( 8-x^2/2) on the interval [0, 4] (a) Find k such that p(x) is a valid PDF. (b) Find P(X ≤ 1). (c) Find the mean, µ, of X. (d) Find the variance, σ 2 , of X.
2. where The random variable, X, has a probability density function (pdf), f(x), and zero otherwise. (a.) (b) x+1, f(x) = {1x, 1 81 -1/2 < x < 0 0
The probability density function of a variate X is as follows, 1 2 3 4 P(X) k 3k 5k 7k 9k 11k 13k Find the value of, (i) k (ii) P(X < 4) (iii) P(3 < X < 6).
6. Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-x, o0) is a function f such that f(r) 20 and (2) = = 1. (a) Determine which of the following functions are probability density functions on the (-x0, 00). fr-1 00 (b) We can also use probability density functions to find the erpected value of the outcomes of the event - if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. rf(x) dr yields the expected value for a density f(r) with domain on the real numbers.) Find the expected value for one of the valid probability densities above.
(b) The probability density of a function is given by p(x) from x = 1 to x = 4. That is 62 x3 dx = 1 62 P(x) dx = - 00 Determine the value of the standard deviation.
A random variable has probability density function f(x) = 30x (1 - x) <(0 < xs 1). Compute its cumulative distribution F(x). O 10x3- 15x4 + 6x5 + 1 O 30x(1 - x) O 30x2 - 60x3 + 30x4 O 10x3 - 15x4 + 6x5 O none of these A Moving to the next question prevents changes to this answer. to search 63°F S End F10 Home %23 PrtScn F8 F2 F3 F4 F5 F6 F7 F9 近

Suppose that buses arrive are scheduled to arrive at a bus stop at noon but are always X minutes late, where X is an exponential random variable with probability density function fx(x) Ae-A. Suppose that you arrive at the bus stop precisely at noon. = (a) Compute the probability that you have to wait for more than five minutes for the bus to arrive. (b) Suppose that you have already waiting for 10 minutes. Compute the probability that you have to wait an additional five minutes or more.
1. A continuous random variable X is defined by (3+x) f(x) - 3sxs-1 %3D 16 (6-2r) -1sxs1 16 (3-x) -1sxs3 16 a. Verify that f(x) is density. Explain b. Find the mean 2. If the probability density function of random variable is given by f(x) = wch 1srs2 sech x a. Find the mean b. Find the total area 3. If the probability density function of random variable is given by f(x) = sinh 2x 1sxs2 a. Find the mean b. Find the total area c. Illustrate the graph (use bell shaped figure) II II
2) The probability density function of a random variable X is giyen by: [kx(2-x),03) 3)
2B. (a) An exponential distribution has probability density function ƒ(x) = ¹²e-x/a, x ≥0. α Show that the mean of the distribution is a and the variance is a². (b) A machine contains two belt drives, of different lengths. These have times to failure which are exponentially distributed, with mean times a and 2a. The machine will stop if either belt fails, and the failures of the belts are independent. Show that the chance that the machine is still operating after a time a from the start is e 32

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Q6. (a) The continuous random variable X has a probability density function given by 2 1 C 2 1< x<2 - f(x)=- x² 2 otherwise where c is a constant. (i) Show that c = 2. (ii) Find P X =