The probability that the day with the least amount of snowfall experiences at most 1.5 inches of snow is approximately 0.528 or 52.8%.
How to calculate the probability
To calculate the probability that the day with the least amount of snowfall experiences at most 1.5 inches of snow, consider the individual probabilities for each day and then determine the minimum of these probabilities.
Let's denote the snowfall on the first day as X₁, the snowfall on the second day as X₂, and the snowfall on the third day as X₃.
We are given that the snowfall on each day follows an exponential distribution with respective means of 1.25, 0.75, and 2 inches.
The exponential distribution has a probability density function (PDF) given by:
f(x) = λ * exp(-λx)
where λ is the rate parameter, given by the reciprocal of the mean (λ = 1/mean).
For the first day, X₁, the mean is 1.25 inches, so the rate parameter is λ₁ = 1/1.25 = 0.8.
Similarly, for the second day, X₂, the rate parameter is λ₂ = 1/0.75 = 1.33.
And for the third day, X₃, the rate parameter is λ₃ = 1/2 = 0.5.
To find the probability that the day with the least amount of snowfall experiences at most 1.5 inches of snow, find the probability that each day's snowfall is at most 1.5 inches and then determine the minimum of these probabilities.
For an exponential distribution with rate parameter λ, the cumulative distribution function (CDF) is given by:
F(x) = 1 - exp(-λx)
So, for each day:
P(X₁ ≤ 1.5) = F(1.5) = 1 - exp(-0.8 * 1.5)
P(X₂ ≤ 1.5) = F(1.5) = 1 - exp(-1.33 * 1.5)
P(X₃ ≤ 1.5) = F(1.5) = 1 - exp(-0.5 * 1.5)
To find the overall probability, we take the minimum of these three probabilities:
P(min(X₁, X₂, X₃) ≤ 1.5) = min(P(X₁ ≤ 1.5), P(X₂ ≤ 1.5), P(X₃ ≤ 1.5))
Calculating each of these probabilities:
P(X₁ ≤ 1.5) = 1 - exp(-0.8 * 1.5)
≈ 1 - exp(-1.2)
≈ 1 - 0.301
≈ 0.699
P(X₂ ≤ 1.5) = 1 - exp(-1.33 * 1.5)
≈ 1 - exp(-1.995)
≈ 1 - 0.135
≈ 0.865
P(X₃ ≤ 1.5) = 1 - exp(-0.5 * 1.5)
≈ 1 - exp(-0.75)
≈ 1 - 0.472
≈ 0.528
Taking the minimum of these probabilities:
P(min(X₁, X₂, X₃) ≤ 1.5) = min(0.699, 0.865, 0.528)
≈ 0.528
Therefore, the probability that the day with the least amount of snowfall experiences at most 1.5 inches of snow is approximately 0.528 or 52.8%.