Question

The value of the expression :
$log_{2}(1 + \frac{1}{2 }{\sum _{k = 1} ^{11} { \: }^{12} c _{k} } )$
is equal to = ?​

Answer 1

Step-by-step explanation:

Problem

Let X,Y and Z be three jointly continuous random variables with joint PDF

fXYZ(x,y,z)=⎧⎩⎨⎪⎪13(x+2y+3z)00≤x,y,z≤1otherwise

Find the joint PDF of X and Y, fXY(x,y).

Solution

Problem

Let X,Y and Z be three independent random variables with X∼N(μ,σ2), and Y,Z∼Uniform(0,2). We also know that

E[X2Y+XYZ]=13,E[XY2+ZX2]=14.

Find μ and σ.

Solution

Problem

Let X1, X2, and X3 be three i.i.d Bernoulli(p) random variables and

Y1=max(X1,X2),Y2=max(X1,X3),Y3=max(X2,X3),Y=Y1+Y2+Y3.

Find EY and Var(Y).

Solution

Problem

Let MX(s) be finite for s∈[−c,c], where c>0. Show that MGF of Y=aX+b is given by

MY(s)=esbMX(as),

and it is finite in [−c|a|,c|a|].

Solution

Problem

Let Z∼N(0,1) Find the MGF of Z. Extend your result to X∼N(μ,σ).

Solution

Problem

Let Y=X1+X2+X3+...+Xn, where Xi's are independent and Xi∼Poisson(λi). Find the distribution of Y.

Solution

Problem

Probability Generating Functions (PGFs): For many important discrete random variables, the range is a subset of {0,1,2,...}. For these random variables it is usually more useful to work with probability generating functions (PGF)s defined as

GX(z)=E[zX]=∑n=0∞P(X=n)zn,

for all z∈R that GX(z) is finite.

Show that GX(z) is always finite for |z|≤1.

Show that if X and Y are independent, then

GX+Y(z)=GX(z)GY(z).

Show that

1k!dkGX(z)dzk|z=0=P(X=k).

Show that

dkGX(z)dzk|z=1=E[X(X−1)(X−2)...(X−k+1)].

Solution

Answer 2

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