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Numerade Educator



Problem 18 Easy Difficulty

Show that the probability density function for a normally distributed random variable has inflection points at $ x = \mu \pm \sigma $.


$\begin{aligned}(-x+\mu)^{2} &=\sigma^{2} \\-x+\mu &=\pm \sigma \\ x &=\mu \pm \sigma \end{aligned}$


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Video Transcript

a normal. Yeah. Normal distribution on this one has set probabilities density function, that is, has two parameters mu the mean on the variance sigma. So these this distribution to the probability density function is she called FX, uh, one over, um, squared off to five them. So, uh, cinema, um sex potential off minus for our X minus. New squared over. Don't, Seema. It's square. So, for these you would like to find a The inflection points off this, uh, no city function. Mhm Pakistan Fortune s O. That means while the inflection points are changes points where the okay, the function changes changes inside off you've uncovered con cavity. Oh, yeah, For diversity on the committee it is, uh, right is directly related toe sign off and rebel Brian. So if we want to change? No, At which point does it change? Can cavity not look at changes in sign a festival frame? Yeah, changes in the sign she would like to find it. What is the secondary fifth for that function? But if prime is gonna be rolled up Constant multiplying there. Yeah, one over scraping five Sigma bonding. The development exponential is the exponential itself. So the exponential about them. Six points old bar function X one squared over to Seema squared on by the channel is gonna be times the derivative of this. But on the exponential that is being exponentially ated so relative of these. Thanks. On the negative of this term is, uh now the power rules is one of u minus. Depends God X minus you to see what square? So at least two this couple off twos cancel. And then the narrative is gonna be a crime is one of the skirt off to fight and Sigma Exponential off minus X minus. Mu squared over two Sigma squared time service. Well, with the miles it was minus X over, it's mine. Eczema X minus moves you over seven squared? No. When we want to differentiate this, uh, even this would be evolved. The constant on touch there. And then it's gonna be the tip of the exponential of these exponential. The same that was this time plus the same exponential of the same from the same term off this time, Uh, times the directive on these second term by the broader group. So if we do that, we obtainable. We have one of us square with Pike and Sigma as we did there. This these over here comes out of the relative of the exponential. So it would be this term again. That was that she would be exponential off nine or six minus brew squared over two Sigma squared again That was this term times minus X minus new over sigma squared And then times this time again minus x his meal over c'mon square. And in that plus, we'll exponential of the same exponential of minus X minus new squared over two sigma squared on the negative of this is just minus one over Cinema Square On the wall note is that for Is this expression just some big factor this term? Because it is common to both off these two. So we ride that out, so it's gonna be the second narrative is exponential off minus X minus Mu squared over to sigma squared that divided by Sigma times The squared up to five on. Then you have these terms quibble. The minus goes away because his minus times minus This is squared over. He's to the fourth power. Yeah, yeah, And then over here we have minus one where sigma squared So well, The initial question. Waas, what is the sign off? F o brien? So when we're looking at this, we know that well, these we will not determine the sign because this is always positive. So the sign is gonna be determined by this expression that we like this and then we'll way have that. That expression makes my sigma We're over explains music or to the overseeing of the four minus. So to find this, where does it change? Sign? Do you want to find serious so well, if you have these two expressions can make a common factor off cinema to the floor. But multiplying on the on the wall, on the top, on the bottom by sigma squared so that this expression would be X minus new squared, no minus sigma squared. Um, that divided by Sigma gene four. When is that equal to zero? So it it would be equal to zero when X minus mu square becomes equal to, um signal square on servo. The solutions to that it has solution involved in the square. So you have X minus new is ableto sigma or another solution is that x minus mu. Physical too minus sigma. Um, so they too solutions are Kerekes signal to move in this right there? Yeah. Sigma plus meals. Yeah, or X is equal toe minus sigma Last new. This is visible in normal. All these initial function itself. Well, you know that. Is there an upside to is something like this? We decided to inflection points, So this Well, um, the sign off. This is not the graph off evolved prime, But the sign off global crime is equivalent to the sign of a polynomial. So the inflection point they're gonna be this points that about one is, uh, mu minus cinema on the other point would be, you know, plus Sigma the collection. Okay, points.