In testing a certain kind of missile, target accuracy is measured by the average distance X at which the missile explodes. The distance X is measured in miles and the sampling distribution of X is given by:

X 0 10 50 100
P(X) 1/40 1?20 1/10 33/40

Calculate the variance of this sampling distribution.

Options:

a) 27.6
b) 5138.7
c) 761.0
d) 253.7
e) 88.0
f) None of the above

Mathematical hope (also known as hope, expected value, population means or simply means) expresses the average value of a random phenomenon and is denoted as E (x). Hope is the sum of the product of the probability of each event by the value of that event. It is then defined as shown in the image, Where x is the value of the event, P the probability of its occurrence, "i" the period in which said event occurs and N the total number of periods or observations.

The variance of a random variable provides an idea of the dispersion of the random variable with respect to its hope. It is then defined as shown in the image.

Then you first calculate E [x] and E [[tex]x^{2}[/tex]], and then be able to calculate the variance.

[tex]E[x]=0*\frac{1}{40} +10*\frac{1}{20} +50*\frac{1}{10} +100*\frac{33}{40}[/tex]

[tex]E[x]=0+\frac{1}{2} +5+\frac{165}{2}[/tex]

E[X]=88

So E[X]²=88²=7744

On the other hand

[tex]E[x^{2} ]=0^{2} *\frac{1}{40} +10^{2} *\frac{1}{20} +50^{2} *\frac{1}{10} +100^{2} *\frac{33}{40}[/tex]

E[x²]=0+5+250+8250

E[x²]=8505

Then the variance will be:

Var[x]=8505-7744

Var[x]=761