probability density function, find SD(X), find cumulative distribution function (CDF)

Question

Let X be the amount of time (years) before a new battery stops working. X has the probability density function

f(t) = { C e^(-0.4t) , t ≥ 0

0 , otherwise,

where C is a constant.

a) Find C (using (Pr(-inf < x < inf) = 1))

b) What is the probability that the battery stops working within the first year?

c) On average how long does it take for a battery to stop working.

d) Compute SD(x)

e) Compute the cumulative distribution of X.

f) Use the CDF from e) to find the probability that X is within two standard deviations of the mean.

Answer 1

a) If we integrate over the entire distribution and set it equal to one, we can solve for C as follows:

-∞ → +∞∫f(t)dt = 1 = 0→∞∫f(t)dt

which just means we can ignore the part where x < 0 since f(x) = 0 there. Solving for C gives

C = 0.4

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b) Integrating the pdf from x = 0 to x = 1 gives a probability of roughly 33%

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c) The formula to find the mean time is

μ_t = -∞ → +∞ ∫ t f(t) dt = 1 / 0.4 = 2.5

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d) The formula to find the standard deviation σ_tσ is

σ_t² = -∞ → +∞ ∫(t - μ_t)² f(t) dt = 6.25

⇒ σ_t = 2.5

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e) The cdf is found by integrating the pdf from 0 to x as follows:

0 → x ∫ f(t) dt = { 1 - e^-0.4t, t > 0

{ 0 , t < 0

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f) The probability that x is within two standard deviations of the mean can be found by integrating the pdf with the lower bound μ_t - 2σ_t = -2.5 to the upper bound μ_t + 2σ_t = 7.5 to find

μ_t - 2σ_t → μ_t + 2σ_t ∫ f(t) dt = 0 → 7.5 ∫ f(t) dt

since the lower bound is less than zero, and the pdf is identically zero when x < 0. The integral evaluates to

95.0213%.

It is good to note that by statistical definitions, the probability that x lies within two standard deviations has a well known value, which is about 95%, so this is consistent with our understanding of statistics.