Suppose that a coin is tossed 100 times. Let X be the number of times that it shows heads. Find the mean and standard deviation of X.
mean = np = (100)(.5) = ?
standard deviation = √npq = √(100)(.5)(.5) = ?
(Note: q = 1 - p)
I'll let you finish the calculations.
To find the mean and standard deviation of X, which represents the number of times a coin shows heads in 100 tosses, we need to use the binomial distribution formula.
The binomial distribution formula is given by:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Here:
n = Number of trials (100 tosses in this case)
k = Number of successful outcomes (number of times coin shows heads, represented by X)
p = Probability of success (probability of getting heads on a single toss, which is 1/2 since there are two equally likely outcomes: heads or tails)
C(n, k) = Number of combinations of choosing k successes out of n trials
Mean of X:
The mean of a binomial distribution is given by the formula:
mean = n * p
In this case, mean = 100 * (1/2) = 50.
Standard Deviation of X:
The standard deviation of a binomial distribution is given by the formula:
standard deviation = sqrt(n * p * (1-p))
In this case, standard deviation = sqrt(100 * (1/2) * (1 - 1/2)) = sqrt(100 * (1/2) * (1/2)) = sqrt(100 * (1/4)) = sqrt(25) = 5.
So, the mean of X is 50 and the standard deviation of X is 5.