Video: Using Probability Density Function of Continuous Random Variable to Evaluate an Unknown

Let 𝑋 be a continuous random variable with probability density function 𝑓(π‘₯) = (4π‘₯ + π‘˜)/21, if 3 ≀ π‘₯ ≀ 4, 𝑓(π‘₯) = 0, otherwise. Find the value of π‘˜.

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Video Transcript

Let 𝑋 be a continuous random variable with probability density function 𝑓 of π‘₯ equals four π‘₯ plus π‘˜ divided by 21 if π‘₯ lies between three and four inclusive and 𝑓 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 π‘˜π‘₯. Therefore, the integral of four π‘₯ plus π‘˜ divided by 21 is two π‘₯ squared plus π‘˜π‘₯ 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, we’re 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 𝑓 of π‘₯ is equal to four π‘₯ plus π‘˜ divided by 21 between three and four, then π‘˜ is equal to seven.

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

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