The strength X of a certain material is such that its distribution is found by X...
The strength X of a certain material is such that its distribution is found by X = e^Y, where Y is N(10, 1). Find the cdf and pdf of X, and compute P(10,000 < X < 20,000). Note: F(x) = P(X ≤ x) = P(e^Y ≤ x) = P(Y ≤ ln(x)) so that the random variable X is said to have a lognormal distribution.
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The strength X of a certain material is such that its
distribution is found by X...
The tensile strength Y of a certain material is such that Y= where X is normally...
The tensile strength Y of a certain material is such that
Y=
where X is normally distributed with a mean of
=10 and a variance of
=1 (Y is said to follow a log-normal distribution. Compute each of
the following:
a. P(Y<7,000) [hint: (lnY-)/~N(0,1)]
b. P(10,000<Y<20,000)
c. P(Y>15,000)
A theoretical justiMication based on a certain material failure mechanism underlies the assumption that ductile strength...
A theoretical justiMication based on a certain material failure mechanism underlies the assumption that ductile strength X of a material has a lognormal distribution. If the parameters are μ=5 and σ=0.1 , Find: (a)μ x and σx (b)P (X >120) (c) P(110 ≤ X ≤ 130) (d)T hemedianductilestrength (e)T heexpectednumberhavingstrengthatleast120,iften different samples of an alloy steel of this type were subjected to a strength test. (f) The minimum acceptable strength, If the smallest 5% of strength values were unacceptable.
X is a positive continuous random variable with density fX(x). Y = ln(X). Find the cumulative...
X is a positive continuous random variable with density fX(x). Y
= ln(X).
Find the cumulative distribution function (cdf) Fy(y) of Y in terms of the cdf of X. Find the probability density function (pdf) fy(y) of Y in terms of the pdf of X. For the remaining problem (problem 3 (3),(4) and (5)), suppose X is a uniform random the interval (0,5). Compute the cdf and pdf of X. Compute the expectation and variance of X. What is Fy(y)?...
X is a random variable with a lognormal distribution and that Y = ln(X) ∼ N(µ,...
X is a random variable with a lognormal distribution and that Y = ln(X) ∼ N(µ, σ2 ). Prove that µX = e ^ (µ+ (σ^2)/2 )
(a) Below is the CDF for a discrete random variable, X if x 1 1/2 if...
(a) Below is the CDF for a discrete random variable, X if x 1 1/2 if 1 x< 2 if 2 x 3 7/8 if 3 x 4 F(x) = 3/4 2 1 if nx <n+1. Describe the probability 2n In general, note that for any positive integer n, F(x) distribution of X by finding P(X 1), P(X = 2), P(X positive integer n, and describe an experiment that would result in this random variable X. 3), and the general...
Question 6 A random variable X has cdf χ20 Plotthe cdf and identif.,(x)-1-0.2~ a) Plot the...
Question 6 A random variable X has cdf χ20 Plotthe cdf and identif.,(x)-1-0.2~ a) Plot the cdf and identify the type of the random variable. b) Find the pdf of X. c) Calculate P[-4eX<-1], P(xS2], P(X=1], Pf2-K6], and P[X>10]. d) Calculate the mean and the variance of X. If the random variable X passes through a system with the following chara cteristic function: e) f) Find the pdf of Y. Calculate the mean and the variance of Y. Good Luck
For a random variable X with cumulative distribution function (cdf) Fx(x) = 1- (2/x)^2 ,x>2. (a).Find...
For a random variable X with cumulative distribution function (cdf) Fx(x) = 1- (2/x)^2 ,x>2. (a).Find the pdf fX(x). (b).Consider the random variable Y = X^2. Find the pdf of Y, fY (y).
(c) In certain problems, we are interested in the proportion of a population that lies between ce...
(c) In certain problems, we are interested in the proportion of a population that lies between certain limits. Such limits are called tolerance limits. Consider the random variable P, which is the proportion of the population that lies between Xa and X(n), the smallest and largest values of a random sample of size n taken from the population. Our goal is to find the PDF of P i. Using the joint PDF of Xa) and X) found in part (b)...
For each of the following functions, (i) find the constant c so that f(x) is a...
For each of the following functions, (i) find the constant c so that f(x) is a pdf of a random variable x, (ii) find the cdf F(x)-P(XSX), (iii) sketch graphs of the pdf f (x) and the distribution function F(x), and (iv) find μ and σ2. (a) f (x) x3/4, 0 <x<c (b) f (x)-(3/16x-,-c < x c
Additional Problem 3. If X is a continuous random variable having cdf F, then its median...
Additional Problem 3. If X is a continuous random variable having cdf F, then its median is defined as that value of m for which F(m) = 0.5. Find the median for random variables with the following density functions (a) f(r)-e*, x > 0 (c) f(x) 6r(1-x), 1. Additional Problem 6. Let X be a continuous random variable with pdf (a) Compute E(X), the mean of X (b) Compute Var(X), the variance of X. (c) Find an expression for Fx(r),...
The tensile strength Y of a certain material is such that
Y=
where X is normally distributed with a mean of
=10 and a variance of
=1 (Y is said to follow a log-normal distribution. Compute each of
the following:
a. P(Y<7,000) [hint: (lnY-)/~N(0,1)]
b. P(10,000<Y<20,000)
c. P(Y>15,000)
X is a positive continuous random variable with density fX(x). Y
= ln(X).
Find the cumulative distribution function (cdf) Fy(y) of Y in terms of the cdf of X. Find the probability density function (pdf) fy(y) of Y in terms of the pdf of X. For the remaining problem (problem 3 (3),(4) and (5)), suppose X is a uniform random the interval (0,5). Compute the cdf and pdf of X. Compute the expectation and variance of X. What is Fy(y)?...
(a) Below is the CDF for a discrete random variable, X if x 1 1/2 if 1 x< 2 if 2 x 3 7/8 if 3 x 4 F(x) = 3/4 2 1 if nx <n+1. Describe the probability 2n In general, note that for any positive integer n, F(x) distribution of X by finding P(X 1), P(X = 2), P(X positive integer n, and describe an experiment that would result in this random variable X. 3), and the general...
Question 6 A random variable X has cdf χ20 Plotthe cdf and identif.,(x)-1-0.2~ a) Plot the cdf and identify the type of the random variable. b) Find the pdf of X. c) Calculate P[-4eX<-1], P(xS2], P(X=1], Pf2-K6], and P[X>10]. d) Calculate the mean and the variance of X. If the random variable X passes through a system with the following chara cteristic function: e) f) Find the pdf of Y. Calculate the mean and the variance of Y. Good Luck
(c) In certain problems, we are interested in the proportion of a population that lies between certain limits. Such limits are called tolerance limits. Consider the random variable P, which is the proportion of the population that lies between Xa and X(n), the smallest and largest values of a random sample of size n taken from the population. Our goal is to find the PDF of P i. Using the joint PDF of Xa) and X) found in part (b)...
For each of the following functions, (i) find the constant c so that f(x) is a pdf of a random variable x, (ii) find the cdf F(x)-P(XSX), (iii) sketch graphs of the pdf f (x) and the distribution function F(x), and (iv) find μ and σ2. (a) f (x) x3/4, 0 <x<c (b) f (x)-(3/16x-,-c < x c
Additional Problem 3. If X is a continuous random variable having cdf F, then its median is defined as that value of m for which F(m) = 0.5. Find the median for random variables with the following density functions (a) f(r)-e*, x > 0 (c) f(x) 6r(1-x), 1. Additional Problem 6. Let X be a continuous random variable with pdf (a) Compute E(X), the mean of X (b) Compute Var(X), the variance of X. (c) Find an expression for Fx(r),...
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