Let Y be a Normal RV with mean 7 and standard devi-
ation 4.
a. Find P (Y = 7).
b. Find P (Y < 5).
c. Find P (Y ≥ 10).
d. Find P (Y < 0).
e. Find E[Y].
To answer these questions, we can use the properties of the normal distribution. In particular, we will use the mean and standard deviation provided.
The probability density function (pdf) of a normal distribution is given by:
f(x) = (1/(σ√(2π))) * e^(-((x-μ)^2 / (2σ^2)))
Where:
μ = mean of the distribution
σ = standard deviation of the distribution
e = base of the natural logarithm (approximately 2.71828)
a. Find P(Y = 7):
Since the normal distribution is continuous, the probability of any specific value is zero. Therefore, P(Y = 7) = 0.
b. Find P(Y < 5):
To find P(Y < 5), we need to calculate the cumulative distribution function (CDF) at 5. The CDF of a normal distribution represents the probability that a random variable is less than or equal to a specific value.
Z = (X - μ) / σ
Plugging in the values we have, Z = (5 - 7) / 4 = -0.5. We can then use a standard normal distribution table or a calculator to find the corresponding probability.
c. Find P(Y ≥ 10):
To find P(Y ≥ 10), we can use the complement rule. P(Y ≥ 10) = 1 - P(Y < 10). We can calculate P(Y < 10) using the CDF as we did in part b, with Z = (10 - 7) / 4 = 0.75.
d. Find P(Y < 0):
To find P(Y < 0), we can again use the CDF. The value of Y cannot be less than 0, since the normal distribution is defined over the entire real line.
e. Find E[Y]:
The expected value (also known as the mean) can be calculated as E[Y] = μ. Therefore, E[Y] = 7.
Note: In practice, there are various methods to calculate probabilities and expected values for normal distributions, including using standard normal tables or statistical software. The method described above is a basic approach using the properties of the normal distribution.