Prob. 3Let X be a random variable with cumulative distribution function (cdf) given by(1-e-x², x ≥ 0={1,-x<0Find the probability that the random variable X falls within one standard deviation of itsmean.Fx (x) =

Let X be a random variable with a cumulative distribution function (cdf) given byTo findThe…

Answered: Prob. 3 Let X be a random variable with…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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(b) Suppose that during rainy season on a tropical island the length of the shower has an exponential distribution, with parameter 1 = 2, time being measured in minutes. What is the probability that a shower will last more than three minutes? If a shower has already lasted for 2 minutes, what is the probability that it will last for at least one more minute?
4. The battery life of a fully charge iPhone 13 Pro on a standby follows a normally distributed with mean 7 hour and variance o² fl = 1 hours. = (a) Write out the integral form of the probability that a fully-charged iPhone 13 Pro last between 7 and 9 hours on standby. then use the pnorm function in R to find this probability. (b) What is the probability that a fully-charged iPhone 13 Pro last less than 5 hours? (c) What is the probability that a fully-charged iPhone 13 Pro last more than 9 hours? (d) What is the minimum time a fully-charged iPhone 13 Pro need to last in order to be in the top 10% best quality?
1 (a) Discuss the two expressions -x;- )² and E(x;- )², both of which are used (n-1) i=1 to measure the spread of a set of observations x1, x2, . .., X. (b) A random sample of n observations is taken from a distribution; the sum of the observations is t, and the sum of the squares of the observations is 12. Explain how to estimate the mean and the variance of the distribution from which the random sample was taken. (c) Given the random sample described in part (b), write down expressions (based on 11 and t2) for estimates of the mean and variance of the mean of a further, independent, random sample of size m , from the original distribution. (d) Given that n = 25, 1, = 400 and t2 = 8800, construct a 99% confidence interval for the mean of the distribution, and use it to test whether or not this mean could be 20.
A random variable X takes value 1 with probability and the value 2 with the remaining probability. What is the variance of X, Var[X]? O 1/3 3/4 2/3 1/2 O 1/4
Suppose that the Cumulative distribution function (CDF) of the random variable X with parameter 0 > 0 is defined by F(x; θ) - 1-el x > 0 i) What is the expected value of X,E(X)? ii) What is the variance of X,V(X)?
2. (a) Let random variables X1 and X2 be independent and have a common distribution of Exponential with mean 1. Find the pdf of Z = X1/X2. (b) What is the name of the distribution of Z?
Suppose a random variable X has the following pdf. The random variable X has a Beta distribution with α = 3 and β = 2. Suppose we wanted to determine Var( 10X - 3 ). By the rules for variances, Var( 10X - 3 ) = * Var(X) + where, using the fact that X has a Beta distribution, Var(X) = Express answer as a decimal. Do not round.
Suppose the lifespan (in months) of a smartphone battery can be modeled as a continuous random variable with CDF F(x) = 1 − e-x/3 x ≥ 0 What is the probability that the battery lasts between 12 to 15 months?
The lifespans of light bulbs are known to follow an exponential distribution. The exponential distribution, often denoted exp(β), is characterized by a single mean parameter β. The exponential distribution has a unique property that its standard deviation is equal to its mean. Suppose you order 100 light bulbs known to each have lifespans (in years) distributed according to exp(4). What is the probability that the average lifespan of your 100 light bulbs exceeds 5 years? (Using R it possible)
I have a probability distribution consisting of (x, p(x)): (-3, 0.1), (-2, 0.2), (-1, 0.3), (0, 0.05). The probabilities obviously do not sum to 1. How do I find <x> (average), <x2> (the second moment), <sigmax2> (the variance)?
X is distributed as a Normal random variable with mean of 100 and a standard deviation of 10 (i.e X~N(100,10). You take a random sample of 20 items from X and you calculate the average of this sample. What is the expected value of this sample average? (i.e. what value would you expect to get for the average of this small sample?
Q7) The length of time, in seconds, that a computer user takes to read his or her e- mail is distributed as lognormal random variable with μ = 1.8 and o² = 4. (a) (b) What is the probability that a user reads e-mail for more than 20 seconds? More than a minute? What is the probability that a user reads e-mail for a length of time that is equal to the mean of the underlying lognormal distribution?

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Prob. 3 Let X be a random variable with cumulative distribution function (cdf) given by Fx(x) = {1 - e-** (1-e-x², x ≥0 x < 0 Find the probability that the random variable X falls within one standard deviation of its mean.