Question

(a) Let X be a continuous random variable which only takes on positive values on the interval [1, 4]. If P(X) = (√ x + √ 1 x )C 2 for all x in this interval, compute the value of C.
(b) Let X be a random variable with normal distribution. Let z represent the z-score for X, and let a be a positive number. Prove that P(z < |a|) = P(z < a) + P(z > −a) − 1.

Answer 1

Explanation:-

Probability density function

Let f(x) be any continuous function of x satisfies the following conditions

i)   

ii)

iii)

Let 'X' be a continuous random variable which only takes on positive values on the interval [1,4]

Given function

Since the total probability is unity

By using integration formula

Now  

now integrating , we get

  

on simplification , we get

On calculation , we get

b)  

Given 'X' be a random variable with Normal distribution and Let Z be the score of X

Hence