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.

Expert Solution

orchestra answered 2 years ago

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.

Assume that a continuous random variable has a following probability density function: f ( x )...

Assume that a continuous random variable has a following probability density function: f ( x ) = { 1 10 x 4 2 ≤ x ≤ 2.414 0 o t h e r w i s e Use this information and answer questions 3a to 3g. Question a: Which of the following is a valid cumulative density function for the defined region ( 2 ≤ x ≤ 2.414)? A.F x ( x ) = 1 50 x 5 −...

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 ) =

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

A continuous random variable X, has the density function f(x) =((6/5)(x^2)) , 0 ≤ x ≤...

A continuous random variable X, has the density function f(x) =((6/5)(x^2)) , 0 ≤ x ≤ 1; (6/5) (2 − x), 1 ≤ x ≤ 2; 0, elsewhere. (a) Verify f(x) is a valid density function. (b) Find P(X > 3 2 ), P(−1 ≤ X ≤ 1). (c) Compute the cumulative distribution function F(x) of X. (d) Compute E(3X − 1), E(X2 + 1) and σX.

2 Consider the probability density function (p.d.f) of a continuous random variable X: f(x) = (...

2 Consider the probability density function (p.d.f) of a continuous random variable X: f(x) = ( k x3 , 0 < x < 1, 0, elsewhere, where k is a constant. (a) Find k. (b) Compute the cumulative distribution function F(x) of X. (c) Evaluate P(0.1 < X < 0.8). (d) Compute µX = E(X) and σX.

The following density function describes a random variable X. f(x)= (x/64) if 0<x<8 and f(x) =...

The following density function describes a random variable X. f(x)= (x/64) if 0<x<8 and f(x) = (16-x)/64 if 8<x<16 A. Find the probability that X lies between 2 and 6. B. Find the probability that X lies between 5 and 12. C. Find the probability that X is less than 11. D. Find the probability that X is greater than 4.

A random variable X has density function f(x) = 4x ( 1 + x2)-3 for x...

A random variable X has density function f(x) = 4x ( 1 + x2)-3 for x > 0. Determine the mode of X.

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...

If a continuous random process has a probability density function f(x) = a + bx, for...

If a continuous random process has a probability density function f(x) = a + bx, for 0 < x < 5, where a and b are constants, and P(X > 3) = 0.3 Determine: The values of a and b. The cumulative distribution function F(x) P(X < 2) P(2 < X < 4)

Question