1. Suppose that X is continuous random variable with PDF f(x) and CDF F(x). (a) Prove...
1. Suppose that X is continuous random variable with PDF f(x) and CDF F(x).
(a) Prove that if f(x) > 0 only on a single (possible infinite) interval of the real numbers then F(x) is a strictly increasing function of x over that interval. [Hint: Try proof by contradiction].
(b) Under the conditions described in part (a), find and identify the distribution of Y = F(x).
2. Suppose now that X ~ Uniform(0, 1). For each of the distributions listed below, use the result from 1.(b) in the previously posted question, to find a function g(x) such that Y = g(x) has the stated distribution.
(a) Exponential (theta)
(b) Uniform (alpha, beta)
(c) Rayleigh (alpha)
(d) Cauchy (alpha, beta)
Homework Answers
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1. Suppose that X is continuous random variable with PDF f(x)
and CDF F(x).
(a) Prove...
Suppose X is an exponential random variable with PDF, fx(x) exp(-x)u(x). Find a transformation, Y g(X) so tha...
Suppose X is an exponential random variable with PDF, fx(x) exp(-x)u(x). Find a transformation, Y g(X) so that the new random variable Y has a Cauchy PDF given 1/π . Hint: Use the results of Exercise 4.44. ) Suppose a random variable has some PDF given by ). Find a function g(x) such that Y g(x) is a uniform random variable over the interval (0, 1). Next, suppose that X is a uniform random variable. Find a function g(x) such...
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given...
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find...
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find the cdf and pdf of |X|. (b) Let Z ~ N(0,1), find the cdf and pdf of |Z| (express the cdf using ” (-), the cdf of Z; give the explicit formula for the pdf).
STAT 115 Let X be a continuous random variable having the CDF Fx(x) = 1 -...
STAT 115 Let X be a continuous random variable having the CDF Fx(x) = 1 - e^ (-e^x) (1) Find the Probability Density Function (PDF) of Y=e^X. (2) Let B have a uniform distribution over (0,1). Find a function G(b) and G(B) has the same distribution as X.
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)?...
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ unifo...
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ uniform(0, 1), find a transformation g) so that X g(U) has the above CDF. (Hint: This entails the quantile function F-().)
2. Suppose that the CDF of X is given by Fur :53 e-3 for x 3....
Consider an exponentially distributed random variable X with pdf f(x) = 2e−2x for x ≥ 0....
Consider an exponentially distributed random variable X with pdf f(x) = 2e−2x for x ≥ 0. Let Y = √X. a. Find the cdf for Y. b. Find the pdf for Y. c. Find E[Y]. If you want to skip a difficult integration by parts, make a substitution and look for a Gamma pdf. d. This Y is actually a commonly used continuous distribution. Can you name it and identify its parameters? e. Suppose that X is exponentially distributed with...
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist...
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X). b) Show that Z=-2ln(Y) has a Gamma dist. & derive it. 4. X_i ~ cont with pdf f_i(x) and CDF F_i(x), i=1, 2, ..., k. all independent. Define Y_i=F_i(X_i), i=1, ..., k. Derive the distribution of U=-2ln[Y_1.Y_2...Y_k].
Problem 2. Assume a random vector (X Y with cdf F(r, ) and pdf f(r,y) (i)...
Problem 2. Assume a random vector (X Y with cdf F(r, ) and pdf f(r,y) (i) Show that Y/X has the pdf f(x, z) |da, g(z) = (ii) For X and identify the distribution of this pdf. xt independent, evaluate the pdf of Y/VX N(0, 1) and Y
(22pts) 6. Suppose X is a continuous random variable with the pdf f(x) is given by...
(22pts) 6. Suppose X is a continuous random variable with the pdf f(x) is given by $(x) = { 1 + 2 OSIS 1; Osasi otherwise. (4 pts) a Verify f(x) is a valid pdf. (4 pts) b. Find the cumulative distribution function (cdt) of X (4 pts) c. Find P(OSX30.5). (5 pts) d. Find E(X). (5 pts) e. Find V(x)
Suppose X is an exponential random variable with PDF, fx(x) exp(-x)u(x). Find a transformation, Y g(X) so that the new random variable Y has a Cauchy PDF given 1/π . Hint: Use the results of Exercise 4.44. ) Suppose a random variable has some PDF given by ). Find a function g(x) such that Y g(x) is a uniform random variable over the interval (0, 1). Next, suppose that X is a uniform random variable. Find a function g(x) such...
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find the cdf and pdf of |X|. (b) Let Z ~ N(0,1), find the cdf and pdf of |Z| (express the cdf using ” (-), the cdf of Z; give the explicit formula for the pdf).
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)?...
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ uniform(0, 1), find a transformation g) so that X g(U) has the above CDF. (Hint: This entails the quantile function F-().)
2. Suppose that the CDF of X is given by Fur :53 e-3 for x 3....
Problem 2. Assume a random vector (X Y with cdf F(r, ) and pdf f(r,y) (i) Show that Y/X has the pdf f(x, z) |da, g(z) = (ii) For X and identify the distribution of this pdf. xt independent, evaluate the pdf of Y/VX N(0, 1) and Y
(22pts) 6. Suppose X is a continuous random variable with the pdf f(x) is given by $(x) = { 1 + 2 OSIS 1; Osasi otherwise. (4 pts) a Verify f(x) is a valid pdf. (4 pts) b. Find the cumulative distribution function (cdt) of X (4 pts) c. Find P(OSX30.5). (5 pts) d. Find E(X). (5 pts) e. Find V(x)
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