Statistics Problem #1

(a) At time t + Δt

P(x successes in time t + Δt)

= P(x successes in time t and 0 success in Δt) +
P(x - 1 successes in time t and 1 success in Δt) +
Σ P(x - k successes in time t and k success in Δt)

f(x, t + Δt) = f(x, t)(1 - λΔt) + f(x - 1, t)λΔt + negligible terms

f(x, t + Δt) - f(x,t) = f(x, t)(λΔt) + f(x - 1, t)λΔt

[f(x, t + Δt) - f(x,t)]/Δt = λ[f(x - 1, t) + f(x, t)]

Let Δt -> 0

df(x,t)/dt = λ[f(x - 1, t) + f(x, t)]

(b) Assume the solution of f(x,t) is (λt)^x exp(-λt)/x!

L.H.S.

= {x(λ^x)[t^(x-1)]exp(-λt)}/x! - [(λt)^x(-λ)exp(-λt)]/x!

= λ{ [(λt)^(x-1)]exp(-λt)/(x - 1)! + [(λt)^x exp(-λt)]/x! }

= λ[f(x - 1, t) + f(x, t)]

= R.H.S.

N.B. You may solve this P.D.E. directly if you are strong in those field.

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