How to proof that Φ(1)=1−Φ(−1)\Phi(1)=1-\Phi(-1) for Standard normal distribution?

I have been studying normal distribution and frequently I see that Φ(1)=1−Φ(−1) where Φ(x)=∫x−∞ϕ(u)du is the cdf of a standard normal variable. How one can prove equality above? and is this true in general? i.e Φ(x)=1−Φ(−x)? Note: I’m pretty beginner with probability so excuse me if this turns out to be very trivial. Thanks Answer … Read more

Large sample confidence interval

Can someone please be so kind to check if my answer is correct Answer See more details in this answer for the definition of a Clopper Pearson confidence interval, and problems with approximations in confidence intervals (the approach you have used). I’ve considered the same principles here. Clopper Pearson confidence interval: with $s=2$ (defects), $n=500$ … Read more

Where do the values in a z-table come from?

I understand the purpose of a z-score and how you calculate z1 and z2, given x1 and x2 for some normal random variable X: Z=x−μσ. If x−μ=σ, then Z=1.00, which tells us that the sample point x is precisely one standard deviation (σ) to the right of its mean (μ). Likewise, if x−μ=−σ, then Z=−1.00, … Read more

Method of Maximum Likelihood for Normal Distribution CDF

Based on a random sample of size n from a normal distribution, X~N(\mu, \sigma^2) find the MLE (maximum likelihood estimator) of the following: P[X>c] for arbitrary c. This seems to be strange question, and the provided solution is even more troubling: Would that not be the method of moments estimate? Surely that solution isn’t correct. … Read more

Converting bounds of double integral to polar coordinates

I’m trying to convert the following to polar coordinates: ∫∞0∫−x−∞12πe−(x2+y2)/2dxdy After converting to polar coordinates, it should be: ∫∞0∫(7/4)π(3/2)π12πe−r2/2drdθ My question is how do we arrive at the bounds of (3/2)π and (7/4)π. Answer The first integral should be ∫∞0dx∫−x−∞12πe−(x2+y2)/2dy wich represent the integral over the half of the 4 quadrant between y axis and … Read more

Evaluating ∫∞0Φ(lnx−μσ)(1−Φ(lnx−μσ))dx\int_0^\infty\Phi\left(\frac{\ln x-\mu}{\sigma}\right)\left(1-\Phi\left(\frac{\ln x-\mu}{\sigma}\right)\right)\mathrm{d}x

I am trying to evaluate the following integral where Φ denotes the CDF of N(0,1): I=∫∞0Φ(lnx−μσ)(1−Φ(lnx−μσ))dx =σeμ∫∞−∞Φ(t)(1−Φ(t))eσtdt The integral arises while computing the Gini coefficient of concentration for the Lognormal distribution. Apparently, the answer should turn out to be (2Φ(σ√2)−1)eμ+σ2/2. I tried integrating by parts taking eσt as the second function but if I do … Read more

Prove that the Distribution of Marginal Vectors are also multivariate normal

I’m told that $\mathbf{\Sigma}$ is a positive definite matrix and also that it’s the variance of $\mathbf{X}$ where $\mathbf{X}=\bf{\mu}+BZ$ such that $\mathbf{\mu},\mathbf{Z}\in\mathbb{R}^n$ and $Z_1,\dots,Z_n\sim_{\mathrm{iid}}\mathcal{N}(0,1)$. In addition, I’m to partition the matrix $\bf\Sigma$ in the following way: $$ \bf \Sigma= \begin{pmatrix} \bf \Sigma_p & \bf \Sigma_r \\ \bf \Sigma_r^{\top} & \bf \Sigma_q \end{pmatrix}. $$ So the … Read more

How do I compute P(Z1>0,Z1+Z2<0)\mathbb{P}(Z_1>0,Z_1+Z_2<0) where Z1Z_1 and Z2Z_2 are independent standard normal variables?

Suppose we have two independent standard normal random variables, Z1,Z2, say. I wanted to calculate this probability P(Z1>0,Z1+Z2<0). I can kind of explain the probability via a diagram—I plotted Z2 against Z1 and said the probability must be a 18 but I wanted a more rigorous approach. Could someone please guide me through how to … Read more

Prove that if X,Y∼N(0,1)X, Y \sim N(0, 1) and α∼U[0,1])\alpha \sim U[0, 1]) are independent, then Xcosα+Ysinα∼N(0,1)X\cos{\alpha} + Y\sin{\alpha} \sim N(0, 1)

Let X,Y∼N(0,1) and α∼U[0,1]), and suppose X,Y,α are independent. Then how to prove that Xcosα+Ysinα∼N(0,1)? For a constant α the claim is obvious but in the case of α∼U[0,1]), the problem appears to be trickier. I think, using the formulas for convolution of probability distributions, the following plan could work: We find the distribution of … Read more

Probability of 0≤X≤Y0\leq X\leq Y for two standard Gaussian random variables random variables

We are given two independent standard Gaussian random variables X∼N(0,σ2x),Y∼N(0,σ2y). Compute Pr(X−Y≤0∩X≥0). Here is what I did so far: Denote by 1(x) a function which is 1 if 0≤x≤y and 0 else. Moreover let F(x) be th CDF of X and G(y) the CDF of Y. Pr(X−Y≤0∩X≥0)=Pr(0≤X≤Y)=∫∫1(x)dF(x)dG(y)=∫∫y0F′(x)dxdG(y)=∫12erf(y√2σ2x)dG(y) But now I don’t know which bound do … Read more