Question

Suppose that the cumulative distribution function of the random variable X is

https://i.imgur.com/taHTeeH.png

apparently the answer to a), b), c), e) are 0.56, 0.7, 0 and 0 respectively.

I have no idea how they got these answer anyone able to help me to understand?

Answer 1

lets draw the distribution graph maybe? or take the derivatives to find the probability density function ....

Answer 2

the derivative of F(x) is simply: f(x) = .25 from 0 to 5

Answer 3

What do I do after I find the f(x)

Answer 4

integrate from x=0 to x=X

Answer 5

plug in the values of x given.

the CDF gives P(X<x)... F(a) = P(X<a) = P(X\(\le\)a)

Answer 6

Okay, so here is the skinny. When it says \(\Pr(X<x)\) or \(\Pr(X\leq x)\), all you gotta do is plug \(x\) into the CDF. For continuous functions like this \(<\) and \(\leq\) are equivalent.

When it says \(\Pr(X>x)\), remember that this is the same as \(1-\Pr(X\leq x)\).

Answer 7

my quandry is that we might need to modify it some to be a true probability function .... since .25 integrated from 0 to 5 is not equal to 1 ... but thats just a thought in my head at the moment

Answer 8

that would just give me 0.25(2.8) - 0

Answer 9

for b and d use the complement...

1-F(a) = P(X>a) = P(X\(\ge\)a)

Answer 10

@Australopithecus Does what I said make any sense?

Answer 11

I think so wio still I have no idea where 0.56 comes from

Answer 12

Actually,your cdf doesn't make any sense... F(5)=P(X<5) = 1.25. Either the cdf is off or the domain is.

Answer 13

it should probably be from 0 to 4.

Answer 14

@pgpilot326 The PDF is \(0.25\). The CDF is \(0.25x\).

Answer 15

yes, and ?

Answer 16

\[F(x)=\begin{cases}0&x<0\\.25x&x:(0,5)\\1&5\le x\end{cases}\]
\[f(x)=\begin{cases}0&x<0\\.25&x:(0,5)\\0&5\le x\end{cases}\]
now the issue is:\[\int_0^{5}.25~dx\ne1\]so this doesnt conform to a proper probability density function to me

Answer 17

so there is something wrong with the question and not my brain?

Answer 18

pfft, others have already stated this while i was trying to figure out the latex :)

Answer 19

yep

Answer 20

This text book trolls me all the time I think

Answer 21

\[
0.20(2.8) = 0.56
\]