Construct the discrete probability distribution for the random variable described. Express the probabilities as simplified fractions. The number of heads in 4 tosses of a coin.

Discrete probability distribution

X           0      1        2       3     4

P(X=x)  1/16  4/16  6/16  4/16  1/16

Step-by-step explanation:

Discrete Random variable:-

A random variable X which can take only a finite number of discrete values

In an interval domain is called a discrete random variable. In other words

a real valued function defined on a discrete sample space is called a

Discrete Random variable.

For example : X(s) = { s: s = 0, 1, 2} The random variable X is a discrete Random variable.

Given the number of heads in 4 tosses of a coin

The total number of possible events is n(S) = 2^4 = 16

Discrete distribution:-

The random variable X is a discrete Random variable.

X(x) = { x: x = 0, 1, 2,3,4}

Here x=0 means → no heads that is the possible outcomes are only one

n(E) = {T,T,T,T} = 1

P(X=0) =

Here x=1 means → one head and three tails that is the possible outcomes are '4'

n(E) ={( HTTT),(THTT),(TTHT),(TTTH)

Here x=2means → TWO head and TWO tails that is the possible outcomes are '6'

n(E) = {( HHTT),(THHT),(TTHH),(HTTH),(THTH),(HTHT)

n(E) = 6

Here x=3means → three head and one tails that is the possible outcomes are '4'

n(E) = { HHHT,HTHH,HHTH,THHH} =3

Here x=4 means → FOUR heads that is the possible outcomes are only one

n(E) = {HHHH} = 1

Final -

Discrete probability distribution

X           0      1        2       3     4

P(X=x)  1/16  4/16  6/16  4/16  1/16

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12