Video Transcript
Let ๐ be a continuous random
variable with probability density function ๐ of ๐ฅ equals one-eighth multiplied by
six ๐ฅ minus seven if ๐ฅ is between two and three inclusively or zero otherwise. Find the probability that ๐ is
between two and 2.5.
A continuous random variable is
characterized by its probability density function. That is a nonnegative function
whose area under the curve is one, and that represents the probability of the whole
sample space. So, we often use integration when
weโre finding a probability from a probability density function of a continuous
random variable. This probability density function
has been given to us as a formula. So, we can use an integral to find
the probability that weโre looking for. That is the probability of ๐ being
between two and 2.5 is the integral between two and 2.5 of ๐ of ๐ฅ with respect to
๐ฅ.
Over the interval that weโre
looking at between two and 2.5, ๐ of ๐ฅ is equal to one-eighth multiplied by six ๐ฅ
minus seven because we were told in the question that this is ๐ of ๐ฅ when ๐ฅ is
between two and three. So, weโre going to find the
integral between two and 2.5 of one-eighth multiplied by six ๐ฅ minus seven with
respect to ๐ฅ. Letโs begin this question by
bringing one-eighth to the front of the integral. So, we now have one-eighth
multiplied by the integral between two and 2.5 of six ๐ฅ minus seven with respect to
๐ฅ. We can now use the usual rules of
integration to integrate six ๐ฅ minus seven.
We know that to integrate a
function of ๐ฅ, we can add one to the power and divide by the new power. And we do this term by term. So, six ๐ฅ integrates to six ๐ฅ
squared over two, but thatโs just the same as three ๐ฅ squared. And negative seven is a constant,
so that integrates to negative seven ๐ฅ. Okay, so weโre nearly there. We just need to apply our limits of
integration. We do this by firstly substituting
in our upper limit of integration, which is 2.5. And then we subtract our integral
with the lower limit substituted, and thatโs two.
So, now, we just need to
simplify. 2.5 squared is 25 over four. So, when we multiply that by three,
we get 75 over four. And seven multiplied by 2.5 is 35
over two. Notice that Iโm keeping everything
in fraction form here. Three multiplied by two squared is
three multiplied by four. So, thatโs 12. And seven multiplied by two gives
us 14.
Okay, to simplify the first
bracket, letโs rewrite 35 over two as something over four. Thatโs going to be 70 over
four. So, simplifying what weโve got, 75
over four minus 70 over four is five over four. So, we have one-eighth multiplied
by five over four. And then subtract this bracket,
which is 12 minus 14. So, thatโs negative two. As negative two is the same as
negative eight over four, we have one-eighth multiplied by five over four minus
negative eight over four. But when we subtract a negative, we
can just add instead. This is one-eighth multiplied by 13
over four. And multiplying the numerators and
the denominators gives us 13 over 32.
So thatโs our final answer. The probability of ๐ being between
two and 2.5 inclusively is 13 over 32.
