Question

1. Suppose a continuous random variable X has the following probability density function egin{equation} f_X(x) = egin{cases} kx(1-x) & 0 le x le 1\ 0 & ext{otherwise} end{cases} end{equation} (a) Find the value of k that makes $f_X(x)$ a valid probability density function. (Recall a property that a PDF must have) (b) Give the CDF, $F_X(x)$. (c) Find $P(0.5 le X le 1)$ using $f_X(x)$. (d) Find $P(0 le X le .75)$ using $F_X(x)$. (e) Find $E(X)$. (f) Find $Var(X)$.

          1. Suppose a continuous random variable X has the following probability density function
egin{equation*}
f_X(x) = egin{cases} kx(1-x) & 0 le x le 1\ 0 & 	ext{otherwise} end{cases}
end{equation*}
(a) Find the value of k that makes $f_X(x)$ a valid probability density function. (Recall a property that
a PDF must have)
(b) Give the CDF, $F_X(x)$.
(c) Find $P(0.5 le X le 1)$ using $f_X(x)$.
(d) Find $P(0 le X le .75)$ using $F_X(x)$.
(e) Find $E(X)$.
(f) Find $Var(X)$.

1. Suppose a continuous random variable X has the following probability density function
eginequation*
fX(x) = egincases kx(1-x) 0 le x le 1 0 extotherwise endcases
endequation*
(a) Find the value of k that makes fX(x) a valid probability density function. (Recall a property that
a PDF must have)
(b) Give the CDF, FX(x).
(c) Find P(0.5 le X le 1) using fX(x).
(d) Find P(0 le X le .75) using FX(x).
(e) Find E(X).
(f) Find Var(X).

Added by Laura A.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

09/15/2023

Step 1

To find the value of k that makes f_x(c) a valid probability density function, we need to ensure that the integral of f_x(c) over its entire domain is equal to 1. Since the function is 0 outside the interval (0, 1), we only need to integrate over that Show more…

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Suppose a continuous random variable X has the following probability density function fX(x) = { kx(1 - x) 0 <= x <= 1; 0 otherwise (a) Find the value of k that makes fX(x) a valid probability density function. (Recall a property that a PDF must have) (b) Give the CDF, FX(x). (c) Find P(0.5 <= X <= 1) using fX(x). (d) Find P(0 <= X <= .75) using FX(x). (e) Find E(X). (f) Find Var(X).