7. A distribution is given with random variables X = 5, 6, 7, 8, 9. The probabilities are as follows; P(5) =0.2, P(6)=0.25, P(7)=0.38, P(8)=0.1, P(9)=0.07. Find the mean, variance, and standard deviation of this distribution.

Q2 It is known that the average mark in a first-year math course exam is normally distributed with a mean of 63 and a standard deviation of 10. Using this information, find 1. The probability that the average exam mark will exceed 75. 2. The probability that the average exam mark will be less than 50. 3. The probability that the average exam mark will lie between 75 and 90. Q3 It is known that the time to failure, in years, of an electronic component is exponentially distributed with λ = 0.2. 1. What is the average time to failure, in years, of this component? 2. Calculate the probability that the time to failure of this component is between 4 and 6 years. 3. What is the expected time to failure of this component if we know it has already lasted 5 years?

Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. Find the 87th percentile.

Q4: Suppose that X has a discrete uniform distribution on the integers 0 through 9. Determine the mean, variance, and standard deviation of the random variable Y = 5X and compare to the corresponding results for X.