Question 1
a. What does it mean for two events A and B to be statistically independent?
b. What is the difference between the standard deviation for Continuous data and the standard deviation for Discrete data (you cannot state that the formula is different)?
Question 2
Digital data are transmitted as a sequence of signals that represent 0s or 1s. Suppose that such data are being transmitted to a satellite and then relayed to a distant site. Suppose that due to electrical interference in the atmosphere, there is a 1-in-1000 chance that the transmitted 0 will be reversed between the sender and the satellite (i.e. distorted to the extent that the satellites receiver interprets the 0 as a 1) and a 2-in-1000 chance that a transmitted 1 will be reversed. Suppose that 40% of the transmitted digits are 0s.
a. What is the probability that a transmitted digit is correctly received by the satellite?
b. Assuming independence, what is the probability that all the digits are received correctly
(i) if 1000 digits are transmitted
(ii) if 10,000 digits are transmitted.
c. Suppose that between the satellite and the receiver, the chances of reversal are twice as large as they were between the sender and the satellite. Assuming independence, what is the probability that a digit reaches the receiver as originally sent?
Question 3
Of 200 adults, 176 own one TV set, 22 own two TV sets, and 2 own three TV sets. A person is chosen at random.
a. What is the probability function of X?
b. What is the expected value for the probability distribution?
c. What are the variance and standard deviations for the probability distribution?
What are you doing Sam?
Question 1:
a. Two events A and B are statistically independent if the occurrence or non-occurrence of one event has no effect on the occurrence or non-occurrence of the other event. In other words, the probability of event A happening does not change based on whether event B happens or not, and vice versa.
To determine if two events A and B are statistically independent, you can use the following formula:
P(A ∩ B) = P(A) * P(B)
If the above equation holds true, then events A and B are independent. If not, they are dependent.
b. The standard deviation for continuous data and the standard deviation for discrete data have the same concept - they both measure the spread or dispersion of the data. However, the calculation of standard deviation can slightly differ depending on the nature of the data.
For continuous data, the standard deviation is typically calculated using integration techniques, such as the formula:
σ = sqrt( ∫ (x - μ)^2 * f(x) dx )
Where σ is the standard deviation, x is the variable, μ is the mean, and f(x) is the probability density function.
For discrete data, the standard deviation is usually calculated using the formula:
σ = sqrt( Σ(x - μ)^2 * P(x) )
Where σ is the standard deviation, x is the value of each data point, μ is the mean, and P(x) is the probability mass function.
While the formulas may look different, the underlying concept of measuring spread remains the same. The main difference is in the calculation technique based on the nature of the data.
Question 2:
a. To find the probability that a transmitted digit is correctly received by the satellite, we need to consider the chances of each digit being reversed.
Since 60% of the transmitted digits are 0s, and there is a 1-in-1000 chance of a 0 being reversed, the probability of a 0 being correctly received is:
P(0 correctly received) = (1 - 1/1000) * 0.6
Similarly, since 40% of the transmitted digits are 1s, and there is a 2-in-1000 chance of a 1 being reversed, the probability of a 1 being correctly received is:
P(1 correctly received) = (1 - 2/1000) * 0.4
b. Assuming independence, the probability that all the digits are received correctly can be found by taking the product of the probabilities calculated in part a.
(i) If 1000 digits are transmitted, the probability that all digits are received correctly is:
P(all digits received correctly) = P(0 correctly received) ^ 600 * P(1 correctly received) ^ 400
(ii) If 10,000 digits are transmitted, the probability that all digits are received correctly is:
P(all digits received correctly) = P(0 correctly received) ^ 6000 * P(1 correctly received) ^ 4000
c. Assuming the chances of reversal between the satellite and the receiver are twice as large, the probabilities calculated in part a need to be adjusted accordingly.
The probability of a digit reaching the receiver as originally sent is:
P(digit reaches receiver intact) = (1 - 2/1000) * P(all digits received correctly)
Question 3:
a. To determine the probability function of X, we need to calculate the probabilities for each outcome of X.
X represents the number of TV sets owned by a randomly chosen person. Based on the given information, we have:
P(X = 1) = 176/200
P(X = 2) = 22/200
P(X = 3) = 2/200
P(X = 0) = 0 (since the person must own at least one TV set)
b. The expected value for the probability distribution can be calculated by multiplying each outcome with its corresponding probability, and then summing them up.
Expected value (E) = (1 * P(X = 1)) + (2 * P(X = 2)) + (3 * P(X = 3))
c. The variance for the probability distribution can be calculated by taking the sum of the squared differences between each outcome and the expected value, weighted by their corresponding probabilities.
Variance = [(1 - E)^2 * P(X = 1)] + [(2 - E)^2 * P(X = 2)] + [(3 - E)^2 * P(X = 3)]
The standard deviation is the square root of the variance.