Does the function f(x)= 2 x / 3 for -1 ≤x ≤2, 0 elsewhere satisfy the two requireme

Does the function f(x)= 2 x / 3 for -1 ≤x ≤2, 0 ...

Key Concepts

Probability Density Function

A probability density function (pdf) is a function that describes the likelihood of a random variable taking on a particular value. It is used in continuous probability distributions, and its primary role is to define a probability distribution for continuous data by assigning probabilities to intervals through integration.

Non-Negativity

A key requirement for a function to be a probability density function is that it must be non-negative for all values in its domain. This means that for every x, the function's value should be greater than or equal to zero, ensuring that probabilities are never negative.

Normalization Condition

Another crucial requirement for a probability density function is that the integral of the function over the entire space must equal 1. This condition, often referred to as the normalization condition, guarantees that the total probability over the entire sample space is 1.

Transcript

00:01 Hi guys, this problem we have f of x is equal to c times 2x minus x a cube.

00:12 This is 4x more than 0 and less than 5 over 2 okay and 0 otherwise.

00:22 So first step we need to rewrite the expression of probability density function 2x minus x cube as negative x times x minus square root of 2 times square root times x plus square root of 2 and here we need to verify whether this function is a probability density function or not so well verify the function for all three cases of c values okay so case 1 if we have c is equal to 0 so in this case, we have integration from negative infinity to infinity for f of x d x is equal to zero.

01:17 But we know that for every probability density function, this should satisfy that integration from negative infinity to infinity for f of x d x must be equal to one.

01:31 So in this case, the f of x is not a probability density function.

01:37 Case 2, if we have c positive more than 0, so in this case we have f of x is negative on the range where x belongs to the interval from negative square root of 2 to 5 over 2.

02:00 Okay so for example fc is equal to two in this case we have f of x is equal to um two times two x minus x cube this is for x minus two x cubed okay so for f of x less than zero we have x belongs to the interval square root of 2 and 5 over 2 that is say x is 1 .5 so f of 1 .5 so it says 4 times 1 .5 minus 2 times 1 .5 power 3 so this is negative 0 .75 and we know that f of x for for any x in the defined interval must be more than 0 and less than 1.

03:16 Okay.

03:18 So in this case, it's not a probability density function on the range of x defined here.

03:25 Okay.

03:27 Case 3, we have fst less than 0.

03:32 And in this case, we have f of x less than 0 on the range from 0 to square root of, 2...

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