1. M

    Showing a family is actually an algebra

    Hello! I am struggling with just the very last stage of the solution and any suggestions would be much appriciated!! Thanks! I am trying to show that given a set X and a collection of subsets U of X, and given that (i) X \in U and \emptyset \in U (ii) E \in U implies that the complement...
  2. Vinod

    find out differential equation of family of circles represented by(x-h)^2+(y-k)^2=a^2

    Differentiating w.r.t.x we have 2x-2+2y dy/dx-2dy/dx =0 So,x - 1+ydy/dx-dy/dx =0 So x-1+(y-1)dy/dx=0 dy/dx=1-x/y-1 differentiating again w.r.t.x,we get 1+(y-1)d²y/dx² +(dy/dx)^2=0 Now how to calculate required differential equation. If my calculations of 1st and 2nd order...
  3. Vinod

    Differential equation of the family of curves y=c(x-c)^2

    Find the differential equation of the family of curves y=c(x-c)^2 where c is a parameter. I don't know the solution to this problem. Shall i get help to get the solution to this problem? Okay,
  4. S

    Randomly chosen family

    A number of children in the families is 0 - 4. Children and probability are: 0 child = 0.15, 1 = 0.25, 2 = 0.3, 3 = 0.2, 4 = 0.1 Question: What is the probability of this randomly chosen family having exactly 2 girls? I think we must consider total...
  5. N

    Family of sets

    The definition of x \in \cap \mathcal{F} is \forall A (A\in \mathcal{F}\Rightarrow x \in A). Similarly, the definition of x \in \cap \mathcal{G} is \forall B (B\in \mathcal{G}\Rightarrow x \in B). Could someone please show me how to define x\in \cap(\mathcal{F} \cup \mathcal{G})?
  6. N

    Family of sets

    \mathcal{F}=\{\{1,2\}, \{3,4\},\{5,6\}\} \mathcal{F} is a family of sets. Question: Is \{\{2,8\}\} a family of set(s)? If not, is there a name for it?
  7. krtica

    A proof for the union of a family of sets..

    What is \bigcup_{n \in N} A_n where A_n=\{n, n+1, n+2,..., 2n\} ? I want to say that the union is \mathbb{N}, could I prove this by induction? What would be the simplest approach?
  8. R

    Conditional probability - family with blue eyes

    Hi all, I'm looking for an answer to the following exercise. I've worked on some but I'm stuck. There is a family with 5 children, and the probability that any child will have blue eyes is 1/4. If you know that at least 1 has blue eyes, what is the probability that 3 or more have blue eyes...
  9. T

    Find 1-parameter family of solutions

    1. (x+2y)dx-(y-1)dy=0 y=-1 x=-2 w=x+2 z=y+1 u=z/w (w+2uw)dw-(uw)(udw+wdu) w(1+2u)dw-w(u)(udw+wdu) (1+2u)dw- (u^2dw+uwdu) (1+2u-u^2)dw- (uw)du dw/w- (u)/(1+2u-u^2)du ln|w|- and here I get a really odd result when I integrate. What am I doing wrong? 2. (3x-2y+4)dx-(2x+7y-1)dy=0
  10. V

    family weight finding problem

    Dave weighs 85 kg. His wife Fiona weighs 15 kg less than he does. Their daughter Kylie weighs half as much as her mother. Their son George weighs 12 kg more than Kylie. How much does the whole family weigh in total? I got 237 kg added together
  11. A

    Bernoulli Family

    A family has 4 children. Assuming that boys and girls are equally likely, calculate the probability, using fractions, and Bernoulli's formula (please use this specific formula when necessary) Probability that: (i) there are exactly 2 boys. (ii) there are exactly 3 girls. (iii) there is at...
  12. H

    Family of Entire Functions

    Problem statement: Let f be an entire function and set \mathscr{F}(f) = \{f(2^n z) : n \in \{0, 1, 2, \dots \} \}. Determine all entire functions f such that \mathscr{F}(f) is a compact normal family on \mathbb{C}. Defs: I'm including these as I'm clear on the definition of normal and...
  13. S

    Multiparameter exponential family

    Let X be distributed as N(\mu, \sigma^2) with n=2 and \theta = (\mu, \sigma) \in R \times R^{+}, where mu and sigma are treated as parameters.. How should I show that this belongs to a two parameter exponential family?
  14. D

    if family equicontinuous, prove lipschitz

    For a function f:R-->R and a>0 set fa(x) = af(x/a). Prove that if the family (fa) a element in (0,infinity) is equicontinuous then f is lipschitz continuous. Okay so i started the proof with a def of equicontinuity and am trying to move towards lipschitz but having trouble
  15. T

    Family of subsets

    Prove: If A \rightarrow B is a function and { C_\lambda \mid \lambda \epsilon \Lambda } is a family of subsets of A, then f ( \bigcup_{\lambda \epsilon \Lambda} C_\lambda) = \bigcup_{\lambda \epsilon \Lambda} f (C_\lambda)
  16. C

    prove F is normal family

    V in C is a region,let z0 in V,and let F={f in H(V):f(z0)=1,Re(f(z))>=0 for all z in V}. Show F is a normal family. I want to use Montel's theorem,but I don't know how to show it is locally bounded.Thanks for any help
  17. O

    Family of subsets

    Prove: If f: A \rightarrow B is a function and { {C_\lambda \mid \lambda \epsilon \Lambda}} is a family of subset of A, then f ( \bigcup_{\lambda \epsilon \Lambda} C_\lambda )= \bigcup_{\lambda \epsilon \Lambda} f(C_\lambda). I not getting family of subsets. How do I start this off. Assume...
  18. K

    normal family, punctured disk

    Let f(z) be analytic on the punctured disk \{ 0 < |z| < 1 \}, and define f_n(z)=f(\frac{z}{n}), n \geq 1. Show that \{ f_n(z) \} is a normal family on the punctured disk if and only if the singularity of f(z) at 0 is removable or a pole. I am having a hard time proving this. I am also not sure...
  19. Danneedshelp

    exponential family help

    Q: Let Y_{1},...,Y_{n} denote a random sample from the probability density function f(y|\theta)=\theta\\y^{\theta\\-1} for 0<y<1, \theta>0 and 0 otherwise. a) Show that this density function is in the (one-parameter) exponential family and that \sum_{i=1}^{n}-ln(Y_{i}) is sufficient for...
  20. O

    nonempty family sets

    Let A be any NONEMPTY family sets. Prove \bigcap_{A \epsilon A} A \subset \bigcup_{A \epsilon A} A .