Prove theorem that says "Convergence in probability implies convergence in distribution" for the case where Xn and X are continuous random variables.
a) Given t and e, show that P(X <= t - e) <= P( Xn <= t + P(|Xn - X| >= e). This gives a lower bound on P(Xn <= t)
b) Use similar strategy to get an upper bound on P(Xn <= t)
c) By pinching, deduce that P(Xn <= t) -> P(X <= t)

what do you mean by fog

step-by-step explanation:

boxplot

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.