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Problem 20 Easy Difficulty

The standard deviation for a random variable with probability density function $ f $ and mean $ \mu $ is defined by
$$ \sigma = \biggl[ \int_{-\infty}^\infty (x - \mu)^2 f(x)\ dx \biggr]^{\frac{1}{2}} $$
Find the standard deviation for an exponential density function with mean $ \mu $.

Answer

$\sigma=\mu$

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Video Transcript

Java UN exponential distribution. Thanks. Exponential distribution. This has the form, the probability distribution off these functions off the form um, Lunda over e la onda today Power. He, uh, didn't lambda minus lambda X for X. You have zero on zero works a smaller. So you want to find them? Standard deviation, that is, uh is defined to be the interval from minus infinity on the infinity, off x minus the mean from the distribution off effects the X square root of that with new demeanor. So all first, want to find out what is the mean for this one? What is that what you knew? And for that we do the mean is equal to she want to integrate X, you know x X. So since we are, that was gonna be equal toe can pull out all things going lambda and goes from zero our community text into the minus Lambda X, the X on the well. For this interval, we can do integration by parts. So here we say tavon in 12 of the farm. You, uh, the V So the in develop you d V is equal toe you them's be my must be means angel off. Really? You? You know well, which is, uh, dvt bt's I need you to be that. So if this is already be, then d is gonna be all toe and minus eat. Today. Mine is Lambda X over London. Because if we differentiate this yet, uh, we get precisely that on board also, that is our B and then our you will be just x. So this integral is gonna be called toe langa. Times be Hugh times v so X from comes that and shall one of Arlanda my seat today my name is Lambda X and then minus so minus minus becomes a plus thesis are the needed in my Islam MDA X over lambda And then x becomes the ex on the, uh well, the Angel. All these would be just me. I'm a slam eggs over minus lambda squared, so it will be equal toe. That directive is this. So it's gonna be that the bottle winter out there on the at the end points, which is gonna be, um, zero on infinity. So, um, e impunity without, uh so And if you think this is gonna go to zero on, that is gonna go zero. So the limit there goes to zero. So the community, the cereal and then at zero is gonna be minus because there is obstructing the Palacio. So mine of that in Islam, the Orlando Soul, one of Rwanda, minus minus one of Rwanda. And then he 20 Hi. So the mean is able to one over Lambda. Oh, so we have that. This is about one of our Lunda, whatever. The minutes one of Orlando. So that we can write our distribution as, uh, off the non zero part he's gonna be, um, he today minus sex over lambda over Mu over here. Yeah, because Islam that it's one of our news And then for the standard deviation you want to compute all the standard deviation of square is going to be equal to the interval from zero to infinity, off X minus mu squared terms the distribution. So the a function the function of the not Lex Onda again, we can integrate by parts integration by parts. So we call this one, um, if you go this hour, the dean on this to be equal toe Well, it's a good day that the TV and then is to be you. So you know, that's our DVD. The is gonna be equal toe minus eat the minus long, the minus X over Mu. If I didn't bite, you squared. Mhm. And then Mm hmm. Um, no. Divided by just Yeah, which is? That would be or the because if we differentiated get that. So the zombie And then well, do you V control Just two times X minus mu. So these interval is gonna be minus eat the minus sex over mule. And then that was that makes money Was meal squared, All right? And then, well, the evaluating the anti derivative so and then minus minus becomes a plus e minus lambda minus x over new do you, do you? Which is that times two x minus. Yeah, And then away into all these, you can do another integration by parts. Yeah, integration by parts. Um, so we go here. Uh, here. This is our the real This is our you. Well, we got we have that as our dividend for please. Let's call the DP Tilda. Tilda. So that is the V. Tilda. The vigil that is equal tool. It reminds us X, when you V Tilda is gonna be able to minus mute them. C minus X overview. Shut up. You have, um, that this whole thing will be will just copying this first part. Pigs. Where were you, Dex? Mines new squared. Then for this part, we will have that minus you. You could have minus six. Overview comes up them to X minus. Mu on. Dana. Ah, well, the house, this our our our do you tell that our YouTube do you kill them is gonna be able to just to for you feel that to be equal to two x minus meal. Yep. That so, then, uh, will be minus on there. Other miners, it becomes, um plus, uh, of the interval off. Two times. Yeah. Mm hmm. To the minus X. Well, this is our p X, then x overview the X, which off the the interval of these is equal through. Yeah, minus two. However, we multiplying already by new coming from there, you know. So you have minus still on the new squared. He today minus Lambda X over meal. So well, the internet. Is that so? It will be this function this function evaluated at the end points. So putting together that information is gonna be or we can factor a B E to the minus six. You? Yeah. Didn't minus sex. You That is multiplying minus X minus. New squared, my six minus new squared. Mm. Minus two X minus. Dream minus two. Thanks. Minus riel. And then Michael, this part might still still can you squared That needs to be evaluated between infinity zero. So Well, this corresponds to deviation. Squared was we have there and so well are impunity your house. The limit on that will be uh huh. Goes to zero The luminess x construction of these sense Oh, minus dot mine two x. My name is Ryu. I was still really squared so that this part goes to zero in the limit. And then there's going to be that minus this computer computer, that, uh, zero. Which is gonna be, um, suit. Huh? Uh, so, let's hear. That's gonna be one on minus this one. Thumbs minus. You squared minus two times, Uh, minus new right minus, uh to, you know, squared. Mm s. Oh, this would be, But there was a new here. So this multiplying their, um you? Yeah. So these movie minus minus mu squared. So this would be plus two new squared. I assume you squared that both of these council eso sigma squared is gonna be equal to minus minus some you squared. So the deviation Physical thio you mean not so that is a creation.