# Probability square of a normal is in a range; mode of gamma

1. If $X \sim N(1,4)$ find $\mathbb{P}(1<X^2<9)$

2. If $x=2$ is the unique mode of the $X \sim \Gamma(2,\beta)$ distribution, find the parameter $\beta$.

Well I tried transformation to solve this problem. Set $Y=X^2$ and I actually found the pdf of $Y$ but when I try to calculate $\mathbb{P}(1<X^2<9)$, I just can't calculate because the pdf of $Y$ is so complicated to integral 1 to 9.

and in number 2. I think I didn't understand the question properly. Does the mode mean that 'the number that came out most'?

I understand the question as like something that if $X\sim \Gamma(r/2,2)$ it follows $X \sim X^2(r)$.

• 1. is find P(1<X-square<9) – tae11 Mar 27 '17 at 6:35
• The $x=2$ is unnecessary, just remove it and the sentence becomes $X \sim \Gamma(2,\beta)$ is the unique mode... which actually makes sense. Now, just look at en.wikipedia.org/wiki/Gamma_distribution, use the $\Gamma(k,\theta)$ definition. BTW, this looks like coursework, and if it is, you have to put in the self-study tag, and ask questions that do not ask for answers, but just help clarifying where you have gotten stuck, which is all I am allowed to answer here. – Carl Mar 27 '17 at 7:35
• @Carl, no, your suggested change doesn't make sense; the original is a little clumsily worded but is clear enough; your change omits something important. I'll make an edit to clarify it. – Glen_b Mar 27 '17 at 8:40
• thanks for your help guys, i use this site first time, and i didn't know how this site goes. anyway i can't see what is the unique thing in X~Gamma(2,Beta), i try to use axioms of probability to find out Beta, but it goes to 1 regardless of Beta value :( – tae11 Mar 27 '17 at 8:43
• @tae11 Your second question was somewhat confusing; I have edited to clarify what I believe was intended. However, as it stands your question doesn't follow our rules. See the help center under homework (whether or not it's literally homework). In particular, you'll need to show what you tried and specifically ask about where you had trouble. – Glen_b Mar 27 '17 at 8:44

1. Don't compute the distribution of $X^2$. Instead (as I already suggested) simply identify all the values of $X$ that satisfy the condition (i.e. work directly with the distribution of $X$). You could do it by computing the distribution of the square but it's effort you don't really need to go to.

2. The mode according to the wikipedia article on it:

The mode of a continuous probability distribution is the value $x$ at which its probability density function has its maximum value.

The density function for a gamma with shape parameter 2 is easy to write down. When finding modes you can safely ignore normalizing constants and work with an unnormalized density, making it easier still.

As I already suggested, figure out where the maximum of that density function is. It's pretty simple -- there's a unique global maximum (which is also the only local maximum and even the only turning point). The density has properties that make this a particularly simple task.

General guidance: in both cases, draw a picture.

• thanks. i got the answer by differentiating the f(x) to find mode in #2. – tae11 Mar 28 '17 at 13:40
• and for #1, i thought about the domain of X, and find out the range for it :) thanks! – tae11 Mar 28 '17 at 13:41