let x be a binomial random variable with expected valuee 4 and variance 2 a) What is P(X=3)?
b) What is P(X<6)?
To answer these questions, we can use the properties of the binomial distribution and the given expected value and variance.
The binomial distribution is characterized by two parameters: n, which represents the number of trials, and p, which represents the probability of success in each trial. In this case, we don't know these values directly, but we can use the relationship between the expected value and variance to find them.
The expected value (E[X]) and variance (Var(X)) of a binomial random variable are given by the formulas:
E[X] = np
Var(X) = np(1-p)
From the given information, we have E[X] = 4 and Var(X) = 2. Substituting these values into the formulas, we get:
4 = np
2 = np(1-p)
From the first equation, we can solve for p:
p = 4/n
Substituting this back into the second equation, we get:
2 = 4(1-4/n)
Simplifying this equation, we get:
1 - 2/n = 1/2
Solving for n, we find:
n = 8
Now that we know the values of n and p, we can proceed to answer the questions.
a) P(X=3) represents the probability of getting exactly 3 successes in the binomial experiment. We can use the probability mass function (PMF) of the binomial distribution to calculate this probability:
P(X=3) = (n choose 3) * p^3 * (1-p)^(n-3)
Substituting n = 8 and p = 4/8 = 1/2, we get:
P(X=3) = (8 choose 3) * (1/2)^3 * (1-1/2)^(8-3)
Calculating this expression, we find:
P(X=3) = 56 * (1/2)^3 * (1/2)^5 = 56 * (1/2)^8 = 0.109375
Therefore, P(X=3) is approximately 0.109375.
b) P(X<6) represents the probability of getting less than 6 successes in the binomial experiment. We can use the cumulative distribution function (CDF) of the binomial distribution to calculate this probability:
P(X<6) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)
We have already calculated P(X=3) in part a). To calculate the other probabilities, we can simply substitute the values of n and p into the PMF formula and calculate them. Summing up all these probabilities, we get:
P(X<6) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)
Calculating this expression, we find:
P(X<6) ≈ 0.890625
Therefore, P(X<6) is approximately 0.890625.