### Video Transcript

Let π be a continuous random variable with probability density function π of π₯ equals four π₯ plus π divided by 21 if π₯ lies between three and four inclusive and π of π₯ equals zero otherwise. Find the value of π.

The integral of any probability density function is equal to one. This means that in our example, integrating four π₯ plus π divided by 21 between the limits three and four will give us an answer of one. Integrating four π₯ gives us two π₯ squared and integrating π gives us ππ₯. Therefore, the integral of four π₯ plus π divided by 21 is two π₯ squared plus ππ₯ divided by 21.

We now need to substitute in the limits four and three and subtract the answers. Substituting in four gives us 32 plus four π divided by 21 and substituting in three gives us 18 plus three π divided by 21. Multiplying each term in this equation by 21 gives us 32 plus four π minus 18 plus three π equals 21.

Grouping the like terms or simplifying the left-hand side of the equation gives us 14 plus π as 32 minus 18 is 14 and four π minus three π is equal to π. Finally, if we subtract 14 from both sides of this equation, weβre left with π is equal to seven as 21 minus 14 is equal to seven. This means that when the probability density function of a continuous random variable is π of π₯ is equal to four π₯ plus π divided by 21 between three and four, then π is equal to seven.

Therefore, π of π₯ is equal to four π₯ plus seven divided by 21 if π₯ lies between three and four and π of π₯ equals zero otherwise.