WEBVTT
00:00:00.600 --> 00:00:16.600
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.
00:00:17.120 --> 00:00:18.280
Find the value of π.
00:00:19.320 --> 00:00:22.840
The integral of any probability density function is equal to one.
00:00:23.400 --> 00:00:33.760
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.
00:00:34.280 --> 00:00:40.160
Integrating four π₯ gives us two π₯ squared and integrating π gives us ππ₯.
00:00:40.800 --> 00:00:48.160
Therefore, the integral of four π₯ plus π divided by 21 is two π₯ squared plus ππ₯ divided by 21.
00:00:49.120 --> 00:00:54.040
We now need to substitute in the limits four and three and subtract the answers.
00:00:54.720 --> 00:01:05.960
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.
00:01:06.640 --> 00:01:17.960
Multiplying each term in this equation by 21 gives us 32 plus four π minus 18 plus three π equals 21.
00:01:18.720 --> 00:01:34.040
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.
00:01:34.640 --> 00:01:45.040
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.
00:01:45.560 --> 00:01:59.240
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.
00:01:59.600 --> 00:02:12.200
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.