A probability density function on R is a function f :R -> R satisfying (i) f(x)≥0...

A probability density function on R is a function f :R -> R satisfying (i) f(x)≥0 or all x e R and (ii) \int_(-\infty )^(\infty ) f(x)dx = 1. For which value(s) of k e R is the function

f(x)= e^(-x^(2))\root(3)(k^(5)) a probability density function? Explain.

Expert Solution

Colby Messinger answered 2 years ago

1. The random variable X has probability density function: f(x) = ( ke−x 0 ≤ x...

1. The random variable X has probability density function: f(x) = ( ke−x 0 ≤ x ≤ ln 5 4 0 otherwise Part a: Determine the value of k. Part b: Find F(x), the cumulative distribution function of X. Part c: Find E[X]. Part d: Find the variance and standard deviation of X. All work must be shown for this question. R-Studio should not be used.

The random variable X has probability density function: f(x) = ke^(−x) 0 ≤ x ≤ ln...

The random variable X has probability density function: f(x) = ke^(−x) 0 ≤ x ≤ ln (5/4) 0 otherwise Part a: Determine the value of k. Part b: Find F(x), the cumulative distribution function of X. Part c: Find E[X]. Part d: Find the variance and standard deviation of X. All work must be shown for this question.

a. For the following probability density function: f(X)= 3/4 (2X-X^2 ) 0 ≤ X ≤ 2 &nbsp

a. For the following probability density function: f(X)= 3/4 (2X-X^2 ) 0 ≤ X ≤ 2 = 0 otherwise find its expectation and variance. b. The two regression lines are 2X - 3Y + 6 = 0 and 4Y – 5X- 8 =0 , compute mean of X and mean of Y. Find correlation coefficient r , estimate y for x =3 and x for y = 3.

Let X has the probability density function (pdf) f(x)={C1, if 0 < x ≤ 1, C2x,...

Let X has the probability density function (pdf) f(x)={C1, if 0 < x ≤ 1, C2x, if1<x≤4, 0, otherwise. Assume that the mean E(X) = 2.57. (a) Find the normalizing constants C1 and C2. (b) Find the cdf of X, FX. (c) Find the variance Var(X) and the 0.28 quantile q0.28 of X. (d)LetY =kX. Find all constants k such that Pr(1<Y <9)=0.035. Hint: express the event {1 < Y < 9} in terms of the random variable X and...

Consider a continuous random variable X with the probability density function f X ( x )...

Consider a continuous random variable X with the probability density function f X ( x ) = x/C , 3 ≤ x ≤ 7, zero elsewhere. Consider Y = g( X ) = 100/(x^2+1). Use cdf approach to find the cdf of Y, FY(y). Hint: F Y ( y ) = P( Y <y ) = P( g( X ) <y ) =

Given the joint probability density function f(x ,y )=k (xy+ 1) for 0<x <1--and--0<y<1 , find...

Given the joint probability density function f(x ,y )=k (xy+ 1) for 0<x <1--and--0<y<1 , find the correlation--ROW p (X,Y) .

Let the continuous random variable X have probability density function f(x) and cumulative distribution function F(x)....

Let the continuous random variable X have probability density function f(x) and cumulative distribution function F(x). Explain the following issues using diagram (Graphs) a) Relationship between f(x) and F(x) for a continuous variable, b) explaining how a uniform random variable can be used to simulate X via the cumulative distribution function of X, or c) explaining the effect of transformation on a discrete and/or continuous random variable

Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R)....

Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R). Explain why {f(x) : f(0) = 1} is not.

Suppose that X has distribution function F (x) and probability densityf(x). Letα̸=0andβ∈R. (i) What is the...

Suppose that X has distribution function F (x) and probability densityf(x). Letα̸=0andβ∈R. (i) What is the distribution function of eX? (ii) What is the distribution function of the random variable αX + β? (iii) What is the probability density function of the random variable αX + β?

Suppose that the distribution of wind velocity, X, is described by the probability density function f(x)...

Suppose that the distribution of wind velocity, X, is described by the probability density function f(x) = (x/σ^2)e^-(x^2/ 2(σ^2)) , x ≥ 0. Suppose that for the distribution of wind velocity in Newcastle, measured in km/hr, σ^2 = 100. (a) In task 1, you showed that the quantile function for this distribution is given by: Q(p) = σ (−2 ln(1 − p))^(1/2), 0 ≤ p < 1 Use this quantile function to generate 100,000 random values from this distribution (when...

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