Which of the following is a valid probability distribution?

The valid probability distribution is:

Probability distribution D.

Probability distribution--

The probability distribution of a discrete variable is the list of the possible value 'x' and the probability of x at one trial.

The probability distribution for a variable x satisfies the following two properties:

                         i.e. 0≤P(x)≤1.

                   i.e. ∑ P(x)=1  .

Now we check which probability distribution satisfies this property:

Probability Distribution A:

        x             P(x)

        1              0.2

        2             0.2

        3             0.2

        4             0.2

        5              0.2

        6              0.2  

                  ∑ P(x)=1.2≠1

Hence, Probability distribution A is not a valid probability distribution.

Probability Distribution B:

        x             P(x)

        1              0.1

        2             0.2

        3             0.3

        4             0.3

        5              0.2

        6              0.1  

                  ∑ P(x)=1.2≠1

Hence, Probability distribution B is not a valid probability distribution.

Probability Distribution C:

        x             P(x)

        1              0.1

        2             0.2

        3             0.4

        4               0

        5              0.1

        6              0.1  

                  ∑ P(x)=0.9≠1

Hence, Probability distribution C is not a valid probability distribution.

Probability Distribution D:

        x             P(x)

        1              0.2

        2             0.1

        3             0.1

        4             0.5

        5               0

        6              0.1  

                  ∑ P(x)=1

Also,

for each x,   0≤P(x)≤1

Hence, Probability distribution D is a valid probability distribution.

if the eqn of the parabola is f(x) = (x-4)(x+2), we need to mult. this out to find the coordinates of the vertex.

then f(x) = x^2 + 2x - 4x - 8, and f(x) = x^2 - 2x - 8.

one way in which you can find the vertex is to calculate x = -b / (2a).   here, that comes to

        )

x = = 1.   f(1) = 1^2 - 2(1) - 8 = - 9.   vertex is at (1, -9).

        2(1) 

find another point on the curve:   let x=0, for which y=-8.   then (0, -8) is on the curve.   plot the vertex and this point and draw a smooth parabola thru both.

hope this