Question

A fair coin is flipped forty times and the number of heads that occur is noted. A random variable X is defined as the number of heads. Find the expected value and the standard deviation.

Answer 1

Answer:

expected value of x  is 1 and standard deviation of x is 0.7071

Step-by-step explanation:

Given data

coin flip = 40 time

heads noted

to find out

the expected value and the standard deviation

Solution

we know coin is flipped 40 times and the number of heads is noted only

so we find standard deviation for x i.e.

standard deviation of  x = $\sqrt{variance x}$   .......1

first we calculate variance x .i.e.

variance x = expected value of x² - (expected value of x)²   .............2

so now we calculate expected value of x

we know when x = 0 , p(x) = 1/4

x = 1 , p(x) = 1/2

x = 2 , p(x) = 1/4

so  expected value of x  =  $\sum_{0}^{2}$ x p(x)

i.e.   expected value of x  =  0 P(x) + 1 p(x) + 2 p(x)

expected value of x  = 0 (1/4) + 1 (1/2) + 2 (1/4)

expected value of x  =  0  +  (1/2) + (1/2)

expected value of x  =    1             ....................3

now calculate expected value of x²

so so  expected value of x²  =  $\sum_{0}^{2}$ x²p(x)

i.e.  expected value of x²  =   x² P(x) + 1² p(x) + 2² p(x)

expected value of x²  =    0 (1/4) + 1 (1/2) + 4 (1/4)

expected value of x² =   0  +  (1/2) + (1)

expected value of x²  = 1.5         .........................4

now put equation 3 and 4 value in equation 2 we get

variance x =  expected value of x² - (expected value of x)²

variance x =  1.5 - (1)²

variance x =  0.5

now put variance value in equation 1 we get

standard deviation of  x = $\sqrt{variance x}$  

standard deviation of  x = $\sqrt{0.5}$  

standard deviation of  x = 0.7071

so standard deviation of x is 0.7071

Answer 2

The expected value is 20 and the standard deviation is 3.16

The expected value

The given parameters are:

n = 40 ---- the number of flips

p = 0.5 --- the probability of obtaining a head

The expected value is calculated as:

$E(x) = np$

This gives

$E(x) = 40 * 0.5$

$E(x) = 20$

Hence, the expected value is 20

The standard deviation

This is calculated as:

$\sigma = \sqrt{np(1 - p)$

So, we have:

$\sigma = \sqrt{40 * 0.5 * (1 - 0.5)$

$\sigma = 3.16$

Hence, the standard deviation is 3.16

Read more about expected value at:

https://brainly.com/question/15858152