Question

In: Statistics and Probability

[25 pts] The probability density function of a random variable X is fX(x) = ce^ -|x|...

[25 pts] The probability density function of a random variable X is fX(x) = ce^ -|x| for all values of x, where c is a constant. Find the value of c and the cumulative distribution function of X.

Solutions

Expert Solution

We are given a random variable X with probability density function given by:

[The support of X is the set of real numbers since the question mentions that for 'all values of x']

Now, we know that the integral of the probability density function of X integrated over the entire support of X must integrate to 1, thus we get:

Now, we find the cumulative distribution function of x.

When x<0, the cumulative distribution function of X is given by:

When x ≥ 0, the cumulative distribution function of X is given by:

Combining equations (1) and (2), we get the cumulative distribution function of X:

For any queries, feel free to comment and ask.

If the solution was helpful to you, don't forget to upvote it by clicking on the 'thumbs up' button.

orchestra answered 1 year ago
Next > < Previous

Related Solutions

1. Let X be a random variable with probability density function fX given by fX(x) =...
1. Let X be a random variable with probability density function fX given by fX(x) = γαγ/ (x + α)^γ+1 , x ≥ 0, 0, x < 0, where α > 0 and γ > 0. (a) Find the cumulative distribution function (cdf) FX of X. (b) Let Y = log(X+α /α) . Find the cdf of Y and identify the distribution. (c) How could a realisation of X be generated from an R(0,1) random number generator? (d) Let Z...
The probability density function of the continuous random variable X is given by fX (x) =...
The probability density function of the continuous random variable X is given by fX (x) = kx, (0 <= x <2) = k (4-x), (2 <= x <4) = 0, (otherwise) 1) Find the value of k 2)Find the mean of m 3)Find the Dispersion σ² 4)Find the value of Cumulative distribution function FX(x)
The probability density function of the random variable X is given by fX(x) = ax +...
The probability density function of the random variable X is given by fX(x) = ax + 2/9 if 1/2 ≤ x ≤ 3, and 0 otherwise. (a) Compute the value of a. (b) Let the random variable Y be defined as Y = [X], where [·] is the “round down” operator (that is, for example, [2.5] = 2, [−2.5] = −3, [−3] = −3). Find the probability mass function of Y . (Hint: For Y to take value k, what...
Let X be a continuous random variable with a probability density function fX (x) = 2xI...
Let X be a continuous random variable with a probability density function fX (x) = 2xI (0,1) (x) and let it be the function´ Y (x) = e −x a. Find the expression for the probability density function fY (y). b. Find the domain of the probability density function fY (y).
If a random variable X has a beta distribution, its probability density function is fX (x)...
If a random variable X has a beta distribution, its probability density function is fX (x) = 1 xα−1(1 − x)β−1 B(α,β) for x between 0 and 1 inclusive. The pdf is zero outside of [0,1]. The B() in the denominator is the beta function, given by beta(a,b) in R. Write your own version of dbeta() using the beta pdf formula given above. Call your function mydbeta(). Your function can be simpler than dbeta(): use only three arguments (x, shape1,...
If a random variable X has a beta distribution, its probability density function is fX (x)...
If a random variable X has a beta distribution, its probability density function is fX (x) = 1 xα−1(1 − x)β−1 B(α,β) for x between 0 and 1 inclusive. The pdf is zero outside of [0,1]. The B() in the denominator is the beta function, given by beta(a,b) in R. Write your own version of dbeta() using the beta pdf formula given above. Call your function mydbeta(). Your function can be simpler than dbeta(): use only three arguments (x, shape1,...
If a random variable X has a beta distribution, its probability density function is fX (x)...
If a random variable X has a beta distribution, its probability density function is fX (x) = 1 xα−1(1 − x)β−1 B(α,β) for x between 0 and 1 inclusive. The pdf is zero outside of [0,1]. The B() in the denominator is the beta function, given by beta(a,b) in R. Write your own version of dbeta() using the beta pdf formula given above. Call your function mydbeta(). Your function can be simpler than dbeta(): use only three arguments (x, shape1,...
The (mixed) random variable X has probability density function (pdf) fX (x) given by: fx(x)=0.5δ(x−3)+ {...
The (mixed) random variable X has probability density function (pdf) fX (x) given by: fx(x)=0.5δ(x−3)+ { c.(4-x2), 0≤x≤2 0, otherwise where c is a constant. (a) Sketch fX (x) and find the constant c. (b) Find P (X > 1). (c) Suppose that somebody tells you {X > 1} occurred. Find the conditional pdf fX|{X>1}(x), the pdf of X given that {X > 1}. (d) Find FX(x), the cumulative distribution function of X. (e) Let Y = X2 . Find...
. Let X and Y be a random variables with the joint probability density function fX,Y...
. Let X and Y be a random variables with the joint probability density function fX,Y (x, y) = { 1, 0 < x, y < 1 0, otherwise } . a. Let W = max(X, Y ) Compute the probability density function of W. b. Let U = min(X, Y ) Compute the probability density function of U. c. Compute the probability density function of X + Y ..
The joint probability density function for two random variables X and Y is given as, fx,y...
The joint probability density function for two random variables X and Y is given as, fx,y (x, y) = (2/3)(1 + 2xy3 ), 0 < x < 1, 0 < y < 1 (a) Find the marginal probability density functions for X and Y . (b) Are X and Y independent? Justify your answer. (c) Show that E[X] = 4/9 and E[Y ] = 7/15 . (d) Calculate Cov(X, Y )
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT