The lifetime of a machine part has a continuous distribution: Osx< 40 fx(x) ={ (10+ x)² else а. Calculate the probability that the lifetime of the machine part is less than 4. b. Calculate the expectation of life of this machine C. Calculate the variance d. Calculate the probability that the lifetime of the machine part is less than 4 given that is lasts for more than 2

Q: (a) Find the expectation E (X) of X. E (x) = [ (b) Find the variance Var(X) of X. Var (x) = 0

A: Consider , the table , x P(x) x*P(x) x^2*P(x) 0 0.4 0 0 1 0.05 0.05 0.05 2 0.15 0.3 0.6 3…

Q: A hobbyist family is making PPE (personal protective equipment) to donate to local health care…

A:

Q: The probability function of the random variable X is defined as f(x)=cx²(1-x)² for 0<x<1, otherwise…

A: Given information: The probability distribution function of the random variable X is defined as:…

Q: Let X = the time between two successive arrivals at the drive-up window of a local bank. If X has an…

A:

Q: Suppose that X and Y have the following joint probability function: 3r, if0<y< a<1 0, f(x, y) =…

A:

Q: (a) Find the expectation E (X) of X. E (X) = [] (b) Find the variance Var(X) of X. Varlx) = D

A: The following table shows the provided outputs of the discrete random variables, along with the…

Q: Let X be a continuous random variable with PDF 2x fx\=) = { otherwise Find the expected value of X.

A:

Q: Assume that Y denotes the number of bacteria per cubic centimeter in a particular liquid and that Y…

A: 3.3 From the given information, Y denotes the number of bacteria per cubic centimeter in a…

Q: The time for the first widget to be manufactured each morning is random between 1 and 3 seconds with…

A: Given density function: fx=1162x-32x+6, 0&lt;x&lt;6 Cumulative distribution function, F(X):…

Q: In a NiCd battery, a fully charged cell is composed of Nickelic Hydrox- yde. Nickel is an element…

A:

Q: Find the probability that the range of a random sample of size 4 from theuniform distribution having…

A: Given: Sample size, n=4 f(x)=1,   0&lt;x&lt;1

Q: Suppose that you have ten lightbulbs, that the lifetime ofeach is independent of all the other…

A: Given: The ten light bulbs are said to be independent of each other based on the lifetime. In which…

Q: Зx (1+x) Q2. If f(x) = ; 0<x <2 denotes the pdf of a continuous random variable, find mean 14 and…

A: From the provided information,

Q: (a) Find the expectation E (X) of X. E (x) = | %3D (b) Find the variance Var(X) of X.

A: The E(X) is defined as E(X) = ∑x.P(X = x) And variance of X is  V(X) = E(X2) - (E(X))2

Q: a) Find the expectation E (X) of X. E (x) = | %3D (b) Find the variance Var(X) of X. Var (x) = 0

A: Here given discrete probability distribution of x  Here use expected value and Variance formula

Q: The observed annual maximum runoff, X, in the Kızılırmak River has been modelled by a normal…

A: Probability of normal distribution:               In a normal distribution, the probability of a…

Q: Let the random variable X have the moment generating function e³t M(t) = 1-t²-1<t<1 What are the…

A:

Q: If the probability density of X is given by f(x) =2x−3 for x &gt; 10 elsewherecheck whether its mean…

A: Solution: From the given information, the probability density function of X is

Q: Consider the Parcto distribution with Cumulative distribution func- tion: (G)". F(X) = 1 Compute its…

A: variance = 3

Q: Verify whether the following function is probability mass function 2x+1 where f(x) 3 x- 0, 1, 2, 3,…

A: A probability mass function gives the probability at each possible value of discrete random…

Q: A researcher finds that the following pdf fits the daily returns for a portfolio manager, f(x|01,…

A:

Q: The diameter of a round rock in a bucket, can be messured in mm and considered a random variable X…

A: We want find  Mean= E(x) , variance = var(x)

Q: a) Find the expectation E (X) of X. E (x) = [] %3D b) Find the variance Var(X) of X. Var(X) Var (X)…

A: Consider the table ,  x P(x) x*P(x) x^2*P(x) 30 0.05 1.5 45 40 0.3 12 480 50 0.1 5 250 60…

Q: 5. Show that for a normal distribution with mean u and variance o', the central moments satisfy the…

A:

Q: Let x be a continuous random variable with PDF 3 x > 1 x4 fx (x) otherwise Find the mean and…

A: Given PDF is fX(x) = 3x4,x ≥ 10,otherwise

Q: 2. The time (in minutes) required for technicians to perform a specific task is believed to follow…

A:

Q: The 3 month demand for a spare car part has the following probability mass function: x 0 1 2 p(x)…

A: Given Data: x 0 1 2 p(x) 0.25 0.5 0.25 a) The expression to calculate the probability of at…

Q: The probability that the percentage of salts in sand from a particular source does not match is 0.1,…

A:

Q: Get the moment-generating function and based on it, calculate the average and variance of X

A:

Q: The ratio of selling prices of a security on consecutive days has a lognormal distribution with…

A: From the information, we have  μ=0.02σ=0.06x=1,5 Therefore, for Lognormal distribution probability…

Q: 5. Suppose that a continuous random variable X has PDF (1 – x²) -1< x<1 f (x) = else (a) Determine…

A:

Q: (a) Find the expectation E (X) of X. E (x) = [] (b) Find the variance Var(X) of X. Var(x) = D %3D

A: Here given discrete probability distribution of X

Q: Suppose Y1, Y2, Y3, and Y4 are mutually independent and identically distributed exponential random…

A: Given  Yi~exp(1)

Q: Q7. A single observation of a r.v. X having an exponential distribution (with mean 0) is used to…

A: Decision rule is given by Ho is accepted if and only if X us less than or equal to 2.

Q: The probability that the percentage of salts in sand from a particular source does not match is 0.1,…

A:

Q: The discrete random variable X has probability mass function kx x = 2,3 2kx P(X = x) = x-1' x = 4,5…

A:

Q: Consider the distribution of Binomial, Poisson, Exponential, Gamma, and let T=X+ 4 & T, = (X- 1).…

A: An estimator is said to be unbiased if E(Statistic)= Parameter If not so  Then the estimator is…

Q: A professor grades on a curve by assigning C's to all scores from µ – ; to µ + %, B’s to all scores…

A: We are given the information that everyone who scores below a D gets an F by the professor and…

Q: Suppose that the unknown X is uniformly distributed between 0 and 2. What is the expected value of…

A:

Q: a) Find the expectation E (X) of X. E(X) = 0 (b) Find the variance VarX) of X. Var(X) = 0

A: Given: x P(x) 0 0.35 10 0.35 20 0.05 30 0.25 Total 1

Q: Let X be a discrete random variable with expectation -0.5 and variance 3. Let Y=−2X+8, find the…

A: The expectation and variance of discrete random variable X are -0.5 and 3, respectively.

Q: The time, X, to infection for Eagle Flu in minutes, after coming into contact with the virus has…

A:

Q: Let X1, X2, ... , Xn be a random sample, normally distributed with mean μ and variance σ2 If σ2 is…

A:

Q: Suppose that the random variable X has the probability density function ´c(1– x²) for -1s x s1 f(x)…

A: f(x)=c(1-x2)     ;  -1≤x≤10                ;  otherwise

Q: Let X1, X2, ..., Xn be a sequence of independent and identically distributed random variables having…

A: Let X1, X2,...,Xn be a sequence of independent and identically distributed random variable having…

Q: Identify the distributions of the random variables with the following moment- generating functions…

A:

Q: Assume the time required to fully recharge an electric car is uniformly distributed between 160 and…

A:

Q: 1. Consider the Gaussian distribution N (m, σ2). (a) Show that the pdf integrates to 1. (b) Show…

A: GivenThe probability distribution function of Gaussian distribution N(m, σ2) is shown…

Q: Find the marginal probability of X = 1.

A: Given:     Y     1 2 3 X 1 0.05 0.05 0 2 0.05 0.1 0.2 3 0.1 0.35 0.1

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!